Gaussian Loop- and Pólya Processes- A Point Process Approach

7/27/2019 Gaussian Loop- and Plya Processes- A Point Process Approach

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Mathias Rafler

Gaussian Loop- and Plya Processes

A Point Process Approach

U n i v e r s i t t P o t s d a m

Universittsverlag Potsdam


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Mathias Rafler

Gaussian Loop- and Plya Processes


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Mathias Rafler

Gaussian Loop- and

Plya Processes

A Point Process Approach

Universittsverlag Potsdam


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Zugl.: Potsdam, Univ., Diss., 2009

Online verffentlicht auf dem Publikationsserver derUniversitt Potsdam:URL http://pub.ub.uni-potsdam.de/volltexte/2009/3870/URN urn:nbn:de:kobv:517-opus-38706http://nbn-resolving.org/urn:nbn:de:kobv:517-opus-38706

Zugleich gedruckt erschienen im Universittsverlag Potsdam:ISBN 978-3-86956-029-8


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Zusammenfassung

Zufallige Punktprozesse beschreiben eine (zufallige) zeitliche Abfolge vonEreignissen oder eine (zufallige) raumliche Anordnung von Objekten. Derenwichtigster Vertreter ist der Poissonproze. Der Poissonproze zum Inten-sitatsma , das Lebesgue-Ma ordnet jedem Gebiet sein Volumen zu,erzeugt lokal, d.h in einem beschrankten Gebiet B, gerade eine mit demVolumen von B poissonverteilte Anzahl von Punkten, die identisch undunabhangig voneinander in B plaziert werden; im Mittel ist diese Anzahl

pB

q. Ersetzt man durch ein Vielfaches a, so wird diese Anzahl mit

dem a-fachen Mittelwert erzeugt. Poissonprozesse, die im gesamten Raumunendlich viele Punkte realisieren, enthalten bereits in einer einzigen Stich-probe genugend Informationen, um Statistik betreiben zu konnen: Bedingtman lokal bzgl. der Anzahl der Teilchen einer Stichprobe, so fragt man nachallen Punktprozessen, die eine solche Beobachtung hatten liefern konnen.Diese sind Limespunktprozesse zu dieser Beobachtung. Kommt mehr alseiner in Frage, spricht man von einem Phasenubergang. Da die Mengedieser Limespunktprozesse konvex ist, fragt man nach deren Extremalpunk-ten, dem Rand.

Im ersten Teil wird ein Poissonproze fur ein physikalisches Teilchen-modell fur Bosonen konstruiert. Dieses erzeugt sogenannte Loops, das sindgeschlossene Polygonzuge, die dadurch charakterisiert sind, da man aneinem Ort mit einem Punkt startet, den mit einem normalverteilten Schritt

lauft und dabei nach einer gegebenen, aber zufalligen Anzahl von Schrit-ten zum Ausgangspunkt zuruckkehrt. Fur verschiedene Beobachtungenvon Stichproben werden zugehorige Limespunktprozesse diskutiert. DieseBeobachtungen umfassen etwa das Zahlen der Loops gema ihrer Lange, dasZahlen der Loops insgesamt, oder das Zahlen der von den Loops gemachten

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Schritte. Jede Wahl zieht eine charakteristische Struktur der invariantenPunktprozesse nach sich. In allen hiesigen Fallen wird ein charakteris-

tischer Phasenubergang gezeigt und Extremalpunkte werden als speziellePoissonprozesse identifiziert. Insbesondere wird gezeigt, wie die Wahl derBeobachtung die Lange der Loops beeinflut.

Geometrische Eigenschaften dieser Poissonprozesse sind der Gegenstanddes zweiten Teils der Arbeit. Die Technik der Palmschen Verteilungeneines Punktprozesses erlaubt es, unter den unendlich vielen Loops einerRealisierung den typischen Loop herauszupicken, dessen Geometrie dannuntersucht wird. Eigenschaften sind unter anderem die euklidische Langeeines Schrittes oder, nimmt man mehrere aufeinander folgende Schritte, dasVolumen des von ihnen definierten Simplex. Weiterhin wird gezeigt, dader Schwerpunkt eines typischen Loops normalverteilt ist mit einer festenVarianz.

Der dritte und letzte Teil befat sich mit der Konstruktion, den Eigen-schaften und der Statistik eines neuartigen Punktprozesses, der PolyascherSummenproze genannt wird. Seine Konstruktion verallgemeinert das Prin-zip der Polyaschen Urne: Im Gegensatz zum Poissonproze, der alle Punkteunabhangig und vor allem identisch verteilt, werden hier die Punkte nach-einander derart verteilt, da der Ort, an dem ein Punkt plaziert wird, eineBelohnung auf die Wahrscheinlichkeit bekommt, nach der nachfolgendePunkte verteilt werden. Auf diese Weise baut der Polyasche Summen-proze Turmchen, indem sich verschiedene Punkte am selben Ort stapeln.Es wird gezeigt, da dennoch grundlegende Eigenschaften mit denjeni-gen des Poissonprozesses ubereinstimmen, dazu gehoren unendliche Teil-barkeit sowie Unabhangigkeit der Zuwachse. Zudem werden sein Laplace-Funktional sowie seine Palmsche Verteilung bestimmt. Letztere zeigt, dadie Hohe der Turmchen gerade geometrisch verteilt ist. Abschlieend wer-

den wiederum Statistiken, nun fur den Summenproze, diskutiert. Je nachArt der Beobachtung von der Stichprobe, etwa Anzahl, Gesamthohe derTurmchen oder beides, gibt es in jedem der drei Falle charakteristischeLimespunktprozesse und es stellt sich heraus, da die zugehorigen Ex-tremalverteilungen wiederum Polyasche Summenprozesse sind.

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Vorwort

Diese Arbeit fat die Ergebnisse meines Studiums der Punktprozesse vonApril 2006 bis Juni 2009 an der Universiat Potsdam unter der Betreuungvon Prof. Dr. S. Rlly und Prof. Dr. H. Zessin (Universitat Biele-feld) zusammen. Dabei war ich seit Oktober 2006 Stipendiat der Inter-

national Research Training Group (IRTG) Stochactic Models of ComplexProcesses Berlin-Zurich (SMCP).

Mein Dank gilt vor allem Prof. Dr. S. Rlly fur die Betreuung, ihr um-fangreiches Engagement, ihre wissenschaftliche Unterstutzung sowie einefortwahrende Bereitschaft zur Diskussion uber den aktuellen Stand der Ar-beit. Des Weiteren mochte ich bei Prof. Dr. H. Zessin bedanken, dessenImpulse und anregende Fragestellungen zu den beiden Themenbereichenfuhrten. Ausgesprochen wertvoll waren ihre zahlreichen, kritischen An-merkungen in der letzten Phase der Arbeit.

Fur die finanzielle Unterstutzung sowie die Forderung bin ich dem IRTGsehr dankbar. Viele Veranstaltungen trugen wesentlich zu einer Erweiter-ung meines Horizonts sowie Einblicken in weitere Gebiete der Wahrschein-lichkeitstheorie bei, dazu mochte ich neben den regelmaigen Seminarenund Minikursen die Fruhlingsschule in Potsdam sowie die Sommerschule inDisentis hervorheben. Das IRTG ermoglichte mir einen erfahrungsreichenAufenthalt an der Universitat Zurich bei Prof. Dr. Bolthausen. Die indiesem Rahmen entstandenen Kontakte haben mich sowohl fachlich als auch

personlich sehr bereichert.Daruber hinaus schatze ich die stets sehr angenehme und offene Atmo-sphare, die sowohl im IRTG, als auch in der Arbeitsgruppe in Potsdamherrschte.

Insbesondere gilt mein Dank meiner Familie, meinen Eltern, die mich

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wahrend des gesamten Studiums sowie der Promotion in jeglicher Hinsichtunterstutzten.

Potsdam, Juni 2009 Mathias Rafler

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Je planmaiger der Mensch vorgeht, um so wirkungsvoller trifft ihn derZufall.

Friedrich Durrenmatt21 Punkte zu den Physikern, Punkt 8


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Contents

0. Introduction 1

I. An Introduction to Point Processes 9

1. Point Processes 11

1.1. Point Processes . . . . . . . . . . . . . . . . . . . . . . . . 12

1.1.1. Basic Notions . . . . . . . . . . . . . . . . . . . . . 12

1.1.2. Moment Measures and Functionals of Point Processes 16

1.1.3. The Poisson Process . . . . . . . . . . . . . . . . . . 18

1.2. The Campbell Measure of a Point Process . . . . . . . . . 231.2.1. Disintegration with respect to the Intensity Measure:

Palm Distributions . . . . . . . . . . . . . . . . . . . 24

1.2.2. Disintegration with respect to the Point Process: Pa-pangelou Kernels . . . . . . . . . . . . . . . . . . . . 26

1.2.3. Construction of Point Processes from Papangelou Ker-nels . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2. Deviation Principles 39

2.1. General Large Deviation Principles . . . . . . . . . . . . . . 40

2.2. Large Deviations for Poisson Processes at increasing Inten-sity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.3. Deviations for Brownian Motions and Brownian Bridges . . 45

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Contents

II. A Bose Gas Model 49

3. Construction of the ideal Bose Gas 513.1. The Loop Space and the Brownian Loop Measure . . . . . . 52

4. Limit theorems and Extremal Measures 63

4.1. The Construction of Martin-Dynkin Boundaries . . . . . . . 66

4.1.1. Local Specifications and Martin-Dynkin Boundary . 66

4.1.2. Counting Loops . . . . . . . . . . . . . . . . . . . . . 67

4.2. The Microcanonical Loop Ensemble . . . . . . . . . . . . . 69

4.3. The Canonical Loop Ensemble . . . . . . . . . . . . . . . . 75

4.4. The Grand Canonical Loop Ensemble . . . . . . . . . . . . 78

4.5. The Canonical Ensemble of Elementary Components . . . 79

5. Geometric Aspects of the ideal Bose Gas 91

5.1. Palm Distributions and Stationarity . . . . . . . . . . . . . 94

5.2. The Barycentre . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.2.1. The Barycentre of a Brownian Loop . . . . . . . . . 98

5.2.2. The Barycentre of a Random Walk Loop . . . . . . 102

5.3. k-Volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.3.1. k-Volumes of independent Vectors . . . . . . . . . . 108

5.3.2. k-Volumes of dependent Vectors . . . . . . . . . . . 110

5.3.3. k-Volumes of Random Walk Loops . . . . . . . . . 112

5.3.4. Rotational invariant Distributions . . . . . . . . . . 115

5.4. Convex Hulls in R2 . . . . . . . . . . . . . . . . . . . . . . 116

5.5. Percolation of Loops . . . . . . . . . . . . . . . . . . . . . . 122

III. A Generalisation of the Polya Urn Schemes: the PolyaSum Process 127

6. The Polya Sum Processes 129

6.1. The Definition of the Polya Sum Process . . . . . . . . . . . 130

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Contents

6.2. Laplace Functionals . . . . . . . . . . . . . . . . . . . . . . 1326.3. Disintegration and Partial Integration . . . . . . . . . . . . 135

7. Limit Theorems for Conditioned Polya Sum Processes 139

7.1. The Turret Ensemble . . . . . . . . . . . . . . . . . . . . . 1407.2. The Brick Ensemble . . . . . . . . . . . . . . . . . . . . . . 1427.3. The General Ensemble . . . . . . . . . . . . . . . . . . . . 1457.4. Large Deviations . . . . . . . . . . . . . . . . . . . . . . . . 147

8. Concluding Remarks 155

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0. Introduction

The accidental occurrence of certain events in time, like incoming phonecalls in a call centre, the growth of a queue, impulses of nerve fibres ordetection of ionising radiation by a Geiger-Muller counter, is a very fun-damental problem in probability. At discrete, but random times specifiedevents occur, which leads to the first idea of counting processes; processeswhich count the number of certain events in some time interval. Such pro-cesses may be easily described by the (random) waiting time between twoevents. An important role play exponential waiting times, since this distri-bution is known to be memoryless. If these waiting times are in additionassumed to be independently and identically distributed, the number ofevents in some given time interval has a Poisson distribution proportionalto the length of the interval. Moreover, the quantities of different periodsof time are independent. These are the characterising properties of thePoisson process.

In focusing the events as points in time, temporal counting processes areextended to spatial counting processes. Events are now points in space,which may be counted in any bounded domain. A primer example is givenby the positions of physical particles, or even individuals, animals, plants,stars. In any case a realisation of a point process is a snapshot of somesituation. Moreover, points may be replaced by geometric objects, such asspheres representing holes in some porous medium or hard-core particles as

well as line segments representing fractures of the earths surface. Naturalquestions refer for instance to the size of clusters built through overlappingobjects. Such percolation problems were addressed by e.g. Hall [Hal85] andwill occur in the second part of this work. In general point processes can bedefined on polish spaces, compact presentations are Kerstan, Matthes and

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0. Introduction

Mecke [KMM74] and Kallenberg [Kal83], Daley and Vere-Jones [DVJ08a,DVJ08b].

Very often earthquakes cause further earthquakes nearby their epicentre,offsprings of a tree grow not too far away from their parental tree or set-tlements are very unlikely to be isolated. The first of the examples maybe considered in time as well, an earthquake causes further earth tremors.These examples show that naturally dependencies between points arise:points affiliate to clusters and define some kind of families of a population.Objects are divided into classes of related objects: the set of points is par-titioned. Considerations about the sizes of families of a population, thoughwithout any spatial or temporal component, can be found in the works ofEwens and Kingman [Kin78a, Kin78b]. Starting from a sample of a pop-ulation of size N, the different species define a (random) integer partitionof N, and by the demand for a consistent sampling procedure, the latterauthor gave a characterisation of the limiting proportions of these specieswhenever the proportions are in descending order. Two interesting aspectsof Kingmans results should be pointed out, and this work shows compara-ble results: Firstly, the set of limits is convex and the limits themselves havea representation as mixtures of extremal points of the whole set. There-fore the analysis may be restricted to the set of extremal points. Secondly,the limiting proportions are not necessarily proportions in the sense thatthey sum up to 1. In fact there is the possibility that really many smallfamilies get lost in the limiting procedure. If the amount of such familiesis sufficiently large, then the small families all together may contribute tothe whole population. Chapter 4, in particular section 4.5 touches relatedquestions.

Permutations on a finite set of N elements define integer partitions bydetermining the cycle sizes of the cycle decomposition of a permutation.

In this manner random permutations lead to random integer partitions. Itis a highly non-trivial task to find out about limiting objects and sizes ofcycles when expanding the set of permuted elements; such questions wereaddressed by Tsilevich [Tsi97], Vershik and Schmidt [VS77, VS78]. Theystudied the asymptotic behaviour of functionals on symmetric groups which

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only depends of the length on the cycles.An important application relating random integer partitions and their

extremal limits on the one hand and point processes on the other handare quantum particle systems. Consider a finite system in equilibrium de-scribed by a Hamiltonian H. A pair

p,

qsatisfying H

, where is

a real number and a square integrable, normalised function with squareintegrable second derivative, represents the system at energy . Such so-lutions satisfy the so-called Boltzmann statistics. Combinatorial difficultiesenter as soon as one demands additional symmetry properties, which areinvariance of under any permutation of the particles,

, for

Bose statistics, and invariance under any permutation with an added minusfor odd permutations, sgnp q , for Fermi-Dirac statistics.

Feynman [Fey48, Fey90] introduced functional integration, which wastreated rigorously by Kac. His method is applied to the object of interest,the statistical operator exp

p H

q, where

0 is the inverse temperature.

Ginibre [Gin71] carried out this analysis and obtained an integration onclosed trajectories, i.e. Brownian bridges, also named loops. For Boltz-mann statistics these loops are exactly of length . The introduction of theinvariance under permutations for the other two statistics is a sophisticatedpart, but has an interesting effect; its treatment was Ginibres importantstep. While for the Boltzmann statistics the end point of each loop is equalto its starting point, in Bose or Fermi-Dirac statistics starting point andend point are not obliged to be identical. Indeed, the symmetrisation of Nelementary trajectories means to obtain the end points of these trajecto-ries from a permutation of their starting points. Since every permutationdecomposes into cycles, the set of elementary trajectories decomposes intoclasses of connected trajectories, where two trajectories and I are con-nected if and only if there exists a sequence of successive trajectories with

the first being and the final one being I

. These classes are called com-posite loops. Therefore in a natural way the equivalence relation on the setof elementary trajectories defines an integer partition of N. One startingpoint of this work will be the interpretation of Ginibres Feynman-Kac rep-resentation of expp Hq as a Poisson process Pz on the space of composite

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0. Introduction

loops.A basic question originates in the pioneering work of Bose and Einstein

in the 1920s about Bosons. They proposed a curious phase transition,nowadays named Bose-Einstein condensation. They showed that if theparticle density exceeds some critical value, a positive fraction of the wholeamount of particles conglomerates or condenses in the lowest eigenstate.In 1938 London proposed that a phase transition between He I and HeII is related to the Bose-Einstein condensation. But not until 1995 Bose-Einstein condensation was observed experimentally in a gas of Rubidiumand Natrium. The physicists Cornell, Ketterle and Wieman received theNobel price for that experiment in 2001. Feynman [Fey53b, Fey53a] againproposed that Bose-Einstein condensation occurs if and only if infinitelylong loops occur with positive probability.

The connection between Bose-Einstein condensation and cycle percola-tion has been established in two articles by Suto [Sut93, Sut02] and Benfattoet al [BCMP05] in the mean field. Suto considers a model on random in-teger partitions and Benfatto et al a mean field model, both did not takespatial relations into account. Fichtner pointed out the connection betweenrandom permutations of countable subsets ofRd and its decomposition intofinite clusters in [Fic91b] and moreover gave a characterisation of the posi-tion distribution of the Bose gas in terms of its moment measures of a pointprocess on Rd in [Fic91a]. Later Ueltschi [Uel06a, Uel06b] examined latticemodels on the basis of Sutos work and thereby introduced so-called spatialpermutations. Very recently, Ueltschi and Betz [BU09, Uel08] generalisedthe lattice model to models of random point configurations in a continuousspace. By symmetrising initial and terminal conditions of Brownian bridgesof a given length , Adams and Konig [AK07] construct for each Brownianbridge a successor starting at the terminal point, and a predator ending at

the starting point. In that way connected bridges define loops (as classesof connected Brownian bridges).In chapters 3 5 a related model is considered. Initial point is the

already mentioned Feynman-Kac representation of the Bose gas obtainedby Ginibre. The specific model is constructed in chapter 3, which contains

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the construction of the space of composite loops and the Poisson processPz thereon. Furthermore properties of the intensity measure z are shown,

such as a factorisation and asymptotics.Chapter 4 is devoted to limit theorems of local specifications derived

from Pz to obtain representation theorems for Pz conditioned on differentinvariant fields. The first section of this chapter introduces the notion oflocal specifications and the Martin-Dynkin boundary technique. Differentways of counting lead to different invariant fields and they are introducedand studied in the following sections. Firstly by counting loops for the mi-crocanonical, canonical and grand canonical ensemble, loop representationsof their corresponding Martin-Dynkin boundary is obtained in terms of ex-tremal points by direct computations. The most delicate part consists ofthe determination of the Martin-Dynkin boundary in the canonical ensem-ble of elementary components in section 4.5. The large deviation principlefrom section 2.2 allows the representation of limits of random integer par-titions in terms of a variational problem with constraints, which is solvedafterwards. This procedures allows the determination of the essential partof the canonical Martin-Dynkin boundary.

The complex of the limits of integer partitions gives insight into a globalproperty with no focus on spatial properties of the loops. Chapter 5 facesgeometric properties of configurations weighted by Pz . The main object isthe typical loop under Pz . Since Pz realises configurations of a countablyinfinite number of loops, and due to the lack of an uniform distribution oncountably infinite sets, there is no natural definition of a typical loop. Thisimplies a change of the point of view on the point process: from the numberof points in some region to the single point. The modern definition of thetypical point has its origin in the work of Kummer and Matthes [KM70]with the introduction of the Campbell measure, which is also developed in

the monograph of Kerstan, Matthes and Mecke [KMM74]. In using thisconcept of the typical loop, properties such as its barycentre, its euclideanlength and the number of its extremal points are considered. Furthermorea percolation problem of the configurations is treated.

A fundamental characteristic of the Poisson process is that points are

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0. Introduction

placed independently and foremost each one with the same intensity. Pa-pangelou processes, apart from the Poisson process, contrast this construc-

tion. In [MWM79] Papangelou processes were characterised by a partialintegration formula. Recently, Zessin [Zes09] gave a direct construction ofthese larger class of processes. Particularly the points are placed accordingto conditional intensities. Zessins construction is reproduced in subsec-tion 1.2.3 and simplified under an additional assumption. These prepara-tions unfold their relevance in chapter 6, where the so-called Polya sumprocess, which firstly occurred in [Zes09], is constructed. Instead of placingthe points independently and, most notably, identically, the mechanism ofplacing the points makes use of Polyas urn dynamics: points are placedsuccessively and each location, at which a point is placed, gets a rewardon the probability to get another point. That way turrets are built frombricks. Apart from this building brick construction, the Polya sum pro-cess is shown to share many properties with the Poisson process, partic-ularly infinite divisibility and complete randomness. Moreover, its Palmkernels are characterised.

In chapter 7 again limit theorems for local specifications are shown, thistime for the Polya sum process and different limiting stochastic fields ob-tained from different observations: firstly by counting turrets, then bycounting bricks and finally by counting turrets and bricks. Particularly themethods used to obtain the last two ensembles are related to the methodsused in section 4.5.

The fundament for these discussions is laid in chapters 1 and 2. Basictools are introduced in required generality and discussed. Section 1.1 dealswith the definition of point processes and Poisson processes on complete,separable metric spaces. Their basic construction via Laplace functionals isrecalled as well as the moment measures defined. Important properties of

the Poisson process, such as complete randomness and infinite divisibility,and their classification into larger classes of point processes follow. Furtherproperties which are not shared by all Poisson processes, like orderliness andstationarity, are reviewed. Particularly stationarity, the invariance undertranslations, leads to helpful factorisations. In section 1.2 the concept of

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the Campbell measure is introduced, and its disintegrations, which lead toPalm and Papangelou kernels, are recalled.

Chapter 2 recalls deviation principles, which are used to obtain proper-ties of the Poisson process Pz constructed in chapter 3 and the limits ofconverging sequences of measures as solution of certain minimisation prob-lems. An important role for the former application play large and smalldeviations of Brownian bridges. They drive the asymptotic behaviour ofthe intensity measure z. The latter application is prepared in section 2.2and applied in chapters 4 and 7.

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Part I.

An Introduction to Point

Processes


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1. Point Processes

In this very first part the theoretical background is developed, the frame-work of point processes on complete, separable metric spaces (c.s.m.s.).This covers the basic objects in subsection 1.1.1, particularly random mea-sures and random point measures following the books of Daley and Vere-

Jones [DVJ08a, DVJ08b] as well as Kerstan et al. [KMM74] and Kallen-berg [Kal83].

Thereafter in subsection 1.1.2 point processes are focused. Besides the in-troduction of the intensity measure and higher moment measures, the mainquestion is to characterise point processes as in von Waldenfels [vW68].Since random measures are non-negative, the Laplace functional turns outto be sufficient. Subsection 1.1.3 starts with the definition of the Poissonprocess on a c.s.m.s. X in terms of its Laplace functional and classifies itsmost important basic properties, which will be needed in this work and un-

derline the nature of the Poisson process. These are complete randomnessgiven by Kingman [Kin67] and infinite divisibility. Further properties likestationarity and orderliness do not follow directly from the general defini-tion of the Poisson process, but need additional assumptions. If X is anAbelian group with corresponding Haar measure (in fact an arbitrary, butfixed Haar measure), then a Poisson process is stationary if and only if itsintensity measure is a multiple of. Even more generally, ifT is an Abeliangroup acting measurably on X, then the factorisation theorem yields that a

T-invariant Poisson processs intensity measure disintegrates into a mul-

tiple m of and a probability measure s on a spce of marks located ats,

pdx

q ms p dx q p dsq .

Moreover the simplicity of a point process is related to properties of the

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1. Point Processes

second order moment measure. The orderliness of a Poisson process alsoaddresses the multiplicity of points. For the Poisson process this shows that

simplicity is equivalent to the absence of atoms of its intensity measure.A fundamental tool in point process theory are Campbell and reduced

Campbell measure recalled in section 1.2, which allows to change the pointof view by disintegration. While the Laplace functional describes the finite-dimensional distributions, the disintegration of the Campbell measure withrespect to the intensity measure of the point process yields the Palm kernel,which is the point process conditioned on the event that at a certain sitea point is present. The result is the point process from the point of viewof a single point which is almost surely present in the realisations. Palmdistributions are the important tool in chapter 5 and occur a second timein chapter 6.

Under additional assumptions, the reduced Campbell measure is abso-lutely continuous with respect to the product of the intensity measure ofand the point process itsself. The disintegration, which yields the Papan-gelou kernel, allows a further change of the point of view: the Papangeloukernel is the intensity of a point conditioned on the presence of a certainconfiguration. These relations were established in [MWM79]. Very recently

Zessin [Zes09] started with some kernel and constructed the point processfor which the kernel is a Papangelou kernel. In section 1.2.3 we reproducehis proof, and give in theorem 1.31 a simpler construction under an addi-tional measurability assumption. Moreover we generalise Zessins proof andcorrect an inaccuracy. A primer example, the Polya sum process definedin [Zes09], is presented and studied in chapters 6 and 7.

1.1. Point Processes

1.1.1. Basic Notions

Let X be a complete separable metric space (c.s.m.s.) and Bp

Xq

the -fieldof its Borel sets, which is the smallest -field containing the open sets. Acontinuous function f : X R therefore is necessarily measurable. Of

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great importance is the ring B0 p Xq of bounded Borel sets allowing us todefine locally finite measures on the measurable space X,B p Xq .Definition 1.1 (Locally finite measures). A Borel measure on the c.s.m.s.X is locally finite if p B q V for every B B0 p Xq .

These measures may contain an infinite mass, but locally only a finiteamount is allowed. Point configurations, i.e. generalised subsets of X,which are locally finite, play a central role. They are expressed as measureswhich only take non-negative integer values on bounded sets.

Definition 1.2 (Measure spaces). Define the following spaces of measures:

i) M p Xq is the space of locally finite Borel measures on B p Xq ,

ii) M p Xq

M

pX

q:

pB

q N d B B0 p Xq

,

iii) M p Xq

M

pX

q: {x}

1

dx

X

,

iv) Mfp Xq

M p Xq : p Xq V

, analogously M fp Xq .

Hence M p Xq is the set of all locally finite point measures on X, M p Xqthe set of all locally finite, simple point measures and likewise Mfp Xq andM fp Xq the corresponding sets of measures of finite total mass. Any locallyfinite subset ofX can be represented as an element ofM p Xq . With M p Xqwe thus allow multiple points.

For measurable functions f : X R write

p fq :

fd.

We say that a sequence of finite measuresp

n q n Mfp Xq converges weaklyif n p fq p fq for any bounded, continuous f : X R. This conceptcarries over to locally finite measures with the additional demand that fhas bounded support.

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1. Point Processes

Definition 1.3 (Vague convergence). A sequence p n q n of locally finitemeasure converges vaguely to if for any continuous f with bounded sup-

port n p fq p fq

For B B p Xq define the evaluation mapping B asB : Mp Xq R { V }, B : p B q . (1.1)

Later B is only considered as a mapping on M p Xq , and therefore takesvalues in N { V }. The next proposition shows their fundamental role,see [DVJ08b, prop. 9.1.IV].

Proposition 1.4. Let X be a c.s.m.s.

i) M p Xq is a c.s.m.s. when endowed with the vague topology.ii) The Borel -algebra BMp Xq is the smallest -algebra on Mp Xq

generated by the mappings {B}B B0 p xq .iii) M p Xq is a c.s.m.s. under the vague topology and its Borel sets agree

with the ones inherited fromM p Xq .In particular the last statement follows directly from the following lemma,

see [DVJ08b, lemma 9.1.V].

Lemma 1.5. M p Xq is a closed subset ofMp Xq .

Let x be the Dirac measure, that is for A B p Xq

x p Aq :

1 if x A

0 if x A.

The next proposition [DVJ08b, prop. 9.1.III] shows that measures M

p Xq decompose into an atomic and a diffuse part, i.e. the former having

non-negative masses on singletons and the latter not. Point measures areparticular examples of purely atomic measures and permit a representationas a sum of Dirac measures. Simple point measures relate locally finitesets of points of X and point measures of X; in fact this correspondence isone-to-one. General point measures M p Xq allow multiple points.

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Proposition 1.6 (Decompositions). Let M p Xq .

i) obeys a unique decomposition into

a

d with a purely atomicpart a, which permits a representation

a i

kixi ,

where the kis are positive real numbers, p xi q i X is an at mostcountable set and d is a diffuse measure.

ii) If M p Xq , then coincides with its atomic part a with addi-tionally the kis being non-negative integers and

p

xiq

i

X has theproperty that p xi q i B is a finite set for any bounded B B0 p Xq .iii) M p Xq if and only if ki 1 for any i.Write x for some M p Xq and x X if p {x}q 0, therefore

p {x}q 1, and say that x is contained in . Thereby any locally finitepoint measure can be represented as

x p {x}q x

with the factors p {x}q 1 if and only if is a simple point measure. iscalled a configuration of elements of X.

The basic terms are defined and point processes may now be defined.

Definition 1.7 (Random measure, Point process).

i) A probability measure on

M

pX

q,B

pX

q

is called a random measure

on Mp Xq ,B p Xq .ii) A probability measure P on M p Xq ,B p Xq is called a point process.iii) A simple point process P is a point process which is concentrated on

M p Xq , PM p Xq 1.

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1. Point Processes

1.1.2. Moment Measures and Functionals of Point Processes

For Borel sets B B p Xq the evaluation mappings B are measurable, hence

random variables, and characteristic values of the evaluation mappings aretheir moments. For a point process P consider the mapping

: B p Xq R , : B Pp B q :

BdP. (1.2)

inherits the finite additivity property and monotone convergence propertyfor increasing sequences Bn B from P and therefore is a measure.

Definition 1.8 (Intensity measure). Let P be a point process. If Mp

Xq , then P is of first order and is called the intensity measure of P.

Also is named first moment measure. Since p B q is the expectation ofB , p B q is the expected number of points of P inside B whether finite ornot. Suppose that the intensity measure exists as a locally finite measure.Let f : X

R be a positive and measurable function, then the randomintegral

f : fp x q p dx qcan be constructed in the usual way as limit of simple functions, i.e. linearcombinations of Bs. Their expectation with respect to a point process Pthen is

Pp fq

fp xq p dxq Pp d q

fp x q p dxq (1.3)

Related integrals will appear in section 1.2. The postponed basic discussionof the so-called Campbell measure will in fact allow the function f to dependon the configuration .

Instead of integrating B, products of the form B1 . . . Bn for notnecessarily disjoint, measurable B1, . . . , Bn may be integrated,

p nq : B p Xq n R , : B1 Bn

B1 BndP.

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p n q can be extended in the usual way from rectangles to general sets in

Bp Xn q , which yields

Definition 1.9 (Higher order moment measures). Let P be a point processand n

N. If p nq M p Xn q , then P is of n-th order and p nq is called then-th order moment measure of P.

Like moments of random variables extend to moment measures, the char-acterisation of non-negative random variables by its Laplace transform car-ries over to the Laplace functional of a point process. Let f : X

R benon-negative, measurable and bounded with bounded support. Then theLaplace functional LP of P is

LP p fq : P

e f

exp

p

fq

P

pd

q.

The importance of the Laplace functional is due to the one-to-one cor-respondence between random measures and functionals which occur asLaplace functionals [DVJ08b, prop. 9.4.II].

Theorem 1.10. Let the functional L be defined for all non-negative, mea-surable and bounded functions f : X R with bounded support. Then L isthe Laplace functional of a random measure P on X if and only if

i) forf1, . . . , f k non-negative, measurable and bounded with bounded sup-port the functional

Lk p f1, . . . , f k; s1, . . . , sk q L

km

1

smfm

is the multivariate Laplace transform of random vector p Y1, . . . , Y k q .

ii) for every sequence fn monotonously converging to f pointwise

Lp

fnq

Lp

fq

iii) L p 0q 1

Moreover, if these conditions are satisfied, the functional L uniquely deter-mines P.

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1. Point Processes

1.1.3. The Poisson Process

Definition 1.11 (Poisson process). Let M p Xq . The Poisson process

with intensity measure is the uniquely determined point process withLaplace transform

P

e f

exp

1 e f

for any continuous, non-negative f with bounded support. Write P for thisprocess.

Putting f

B with B B1 . . . Bk for pairwise disjoint and bounded

B1

, . . . , Bk, we get that the family B1 , . . . , Bk is mutually independentwith each Bm having Poisson distribution with intensity p Bm q . The in-dependence property of P is known as complete randomness, in particu-lar Poisson processes are prime examples for completely random measures.An important representation theorem was given by Kingman [Kin67], alsoin [DVJ08b, thm. 10.1.III].

Theorem 1.12 (Kingman). The log-Laplace functional of a completely ran-dom measure is of the form

log LP p fq p fq Vk

1

kfp xk q 1 e ufp xq p dx, du q , (1.4)where M p Xq is a fixed, non-atomic measure, p xk q k enumerates anat most countable, locally finite set of atoms of P, p k q k is a family oflog-Laplace transforms of positive random variables and is the intensitymeasure of a Poisson process on X R

which satisfies for every 0 theintegrability conditions for each B B0 p Xq ,

p B, p , V q q V 0

u p B, duq V .

Conversely, each log-Laplace functional satisfying equation (1.4) defines acompletely random measure.

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Thus completely random measures may consist of a fixed, non-atomicpart , an atomic part at the sites p xk q k with random weights and a com-

pound Poisson part independent of the atomic part. For a completely ran-dom measure to be a point process, needs to vanish, each k needs to bethe log-Laplace transform of an integer-valued random variable and

pB

q

needs to be a measure on the positive integers. Additionally, to be Poisson,k p sq uk p 1 e

sq and d 1. The intensity measure of a Poisson

process P then is d

k ukxk , which is exactly the decompositionin theorem 1.6 of into its diffuse and atomic part, respectively.

Sums of independently Poisson distributed random variables again havea Poisson distribution, and the latter intensity is exactly the sum of the for-mer ones. Vice versa, each Poisson random variable can be represented as asum of any given number of independently, identically Poisson distributedrandom variables. This property is known as infinite divisibility. Conse-quently, a point process, or more general a random measure, is infinitelydivisible, if it can be represented as a superposition of k independent, iden-tically distributed point processes or random measures for any k. For aPoisson process with intensity choose k for a given non-negative integerk to obtain the representation of the infinitely divisible Poisson process, seee.g. [DVJ08b, thm. 10.2.IX].

Theorem 1.13 (Levy-Khinchin representation). A random measure P isinfinitely divisible if and only if its log-Laplace functional permits a repre-sentation

log LP p fq p fq

1

exp

p

fq

pd

q(1.5)

where Mp Xq and is a -finite measure onM p Xq z {r } such that forevery B B0 p Xq the integrability condition

R

1 e u 1B p duq Vis satisfied. Conversely, such measures and define via equation (1.5)an infinitely divisible random measure.

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1. Point Processes

In the point process case, necessarily 0 and is concentrated on

M

p Xq . Such infinitely divisible random measures play an important role

in limit theorems.Now assume that in addition X is an Abelian group with the commutative

group operation

. There exists a measure M

pX

qwhich is invariant

under the group action, i.e. p A x q p Aq for every A B p Xq , x Xand A x : {a x : a A}. is called Haar measure and is uniquelydetermined up to a positive constant, see e.g. Stroppel [Str06, thm. 12.23].Every x

X induces an automorphism Tx on M p Xq by

Tx p B q : p B xq .Because of Tx

y TxTy, the set of automorphisms T {Tx}x

X is anAbelian group itself and in a natural way homeomorphic to X. A pointprocess P is T-invariant, if the action ofT conserves the distribution,

Pp TxA q Pp Aq

for every x X. In case ofT being a translation group, P is also calledstationary.

Since a point process P is determined by its Laplace functional LP, LPmust be stationary itself. Particularly the log-Laplace functional of a Pois-son process satisfies

log LP

e fp xq

1

e fp xq

1

e f

and since the measure is uniquely determined by the set of continuousfunctions and is stationary, can only be a multiple of the Haar measure on

X,B p Xq

. Therefore a Poisson process is stationary if and only if its

intensity measure is a multiple of the Haar measure on X.More generally, T{Ts}s

G can be an Abelian group defining trans-formations on X, where G is a complete, separable metric group, which islocally compact. At least partial results carry over to this more general caseand a factorisation theorem 1.14 below states that a T-invariant measure

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decomposes into a measure which is a multiple of a Haar measure on G anda second measure on some other space, see e.g. [DVJ08a, prop. A2.7.III].

Proposition 1.14 (Factorisation). Let X be a c.s.m.s, T {Ts}s

G acomplete, separable, locally compact metric group of transformations actingmeasurably on X. Furthermore suppose that there exists a one-to-one, bothways measurable and bounded sets conserving mapping : G Y X withsome c.s.m.s. Y, which preserves the shiftsTg in the sense thatTg p h, y q

pg

h, y

q . Then anyT-invariant measure Mp Xq can be representedas

p fq Y

G

f

p g, y q

p dg q p dy q ,

where is the Haar measure and is up to a constant a unique measureon Y for measurable, non-negative functions f.

Such a situation will occur in section 3.1, where a group of translationsacts on a space of functions and a translation invariant measure is decom-

posed into such two parts. A further application will be in section 5.1the disintegration of the so-called Campbell measure of a stationary pointprocess with respect to its intensity measure.

An important question are criteria for the simplicity of point processes,particularly of Poisson processes. A first characterisation involves the sec-ond order moment measure p 2 q of a point process P, see e.g. [DVJ08b,prop. 9.5.II]. Let the diagonal of a set A B p Xq be

diag Ak

: {p x1, . . . , xk q X

k

: x1

. . .

xk

A}.

Proposition 1.15. A point processP of second order satisfiesp 2 q p diag B2 q p B q for all B B0 p Xq with equality if and only if P is simple.

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1. Point Processes

Since the Poisson process P is completely random,

p

2q

p B1 B2 q B1B2dP p B1 z B2 B1 B2 q p B2 z B2 B1 B2 q dP p B1 z B2 q p B2 z B1 q p B1 B2 q

p B1 z B2 q p B2 z B1 q

B1 B2 p B1 B2 q

2dP p B1 B2 q

2

pB1 q p B2 q

B1 B2 p B1 B2 q

2dP

p B1 q p B2 q p B1 B2 q .

Hence the second order moment measure of P for a product is the productof the intensities plus the variance on the common part of these two sets.Particularly on products of disjoint sets the last term vanishes.

The following concept of orderliness also addresses the multiplicity ofpoints of a point process and particularly for Poisson processes. A pointprocess P is orderly, if the probability of finding many points in a sequenceof shrinking spheres vanishes sufficiently fast compared to the probabilityof finding some point,

PS p xq 1 oPS p xq 0 as 0 for every x X,where S p x q is the sphere with centre x and radius . For a Poisson processP

P

S p xq 1

P

S p xq 0

1 e S p xq

S p xq

e S p xq

1 e

S p xq

vanishes as 0 if and only if x is not an atom of , that is p {x}q 0.Therefore, see [DVJ08b, thm. 2.4.II],

Theorem 1.16. A Poisson processP is orderly if and only if P is a simplepoint process.

Even more, orderliness and complete randomness together characterisePoisson processes, see [DVJ08b, thm. 2.4.V]:

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1.2. The Campbell Measure of a Point Process

Theorem 1.17. A point process P without fixed atoms is a Poisson processif and only if P is orderly and completely random.

Directly from the definition of the Poisson process and the subsequentdiscussion we get a local representation of a Poisson process P, i.e. if isEB-measurable for some B B0 p Xq , a Poisson process P can be written as

P,B p q e p B q

Vn

0

1

n!

Bn

p x1 . . . xn q p dx1 q . . . p dxn q ,

hence can be interpreted as first choosing an integer n according to a Pois-son distribution with intensity and then placing n points independentlyaccording to in .


As mentioned, the integral in equation (1.3) can be extended to functions fdepending on x and additionally on the configuration . The basic step isto attach to each point x of a configuration the configuration itself. Thisoperation is checked to be measurable: set

C p x, q X M p Xq : p {x}q 0,then according to [KMM74, prop. 2.5.1], C is B p Xq BM p Xq -measur-able, and [KMM74, thm. 2.5.2] states

Theorem and Definition 1.18 (Campbell measure). For any integrableor non-negative function h : X M p Xq R the mapping

hp x, q p dx q

is measurable and the Campbell measure CP of a point process P on X isgiven by

CP p hq

hp x, q p dxq Pp d q .

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1. Point Processes

A characterisation of measures which may occur as a Campbell measureof a random measure is given by Wegmann [Weg77]. However, such a

characterisation is not needed in this work.A special case of such integrals occurred in section 1.1.2 as the expec-

tation of random integrals h with h only depending on x. Here even thedependence on the whole configuration is allowed. Recall also that in choos-ing hA p x, q 1A p xq , CP p h

p q

q reduces to the intensity measure of the pointprocess P. Hence the Campbell measure is an extension of the intensitymeasure.

Closely related is the reduced Campbell measure C!P of a point processP. Instead of attaching the whole configuration to x , is reducedbeforehand by x, i.e. x is attached to x.

Definition 1.19 (Reduced Campbell measure). The reduced Campbell mea-sure C!P of a point process P on X is the measure on X M p Xq givenby

C!P p hq

hp x, x q p dxq Pp d q , h 0 measurable.

1.2.1. Disintegration with respect to the Intensity Measure:Palm Distributions

Campbell measure and reduced Campbell measure gain their importancedue to two basic disintegrations, which are now going to be explored. Thenext proposition [DVJ08b, prop. 13.1.IV] demonstrates that for each A

B p M p Xq q the Campbell measure CP p A q ofP is absolutely continuouswith respect to the intensity measure ofP. Its Radon-Nikodym derivativethen is the measurable function Px which is uniquely determined up to sets

of -measure zero.

Proposition 1.20 (Disintegration). Let P be a point process with finiteintensity measure . Then there exists a Palm kernel, a regular family oflocal Palm measures {Px}x

X, which is uniquely defined up to -null sets

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and

CP p hq hp x, q Px p d q p dxqfor non-negative or CP-integrable h.

The disintegration result of the Campbell measure with respect to theintensity measure leads to the interpretation that Px is the original processP conditioned on the event that there is at least a point at x, i.e. condi-tioned on the event

{x} 0

. In the special case of the Poisson process

these Palm kernels take a simple form and moreover characterise the Pois-son process uniquely. Similar characterisations can be shown for a larger

class of processes.Theorem 1.21 (Meckes characterisation of the Poisson process). There isexactly one point process P satisfying for any measurable, non-negative h

CP p hq

h

px,

x q Pp d q p dxq . (1.6)

P

P is the Poisson process with intensity measure .

In replacing CP by the reduced Campbell measure C!P, equation (1.6) is

equivalent to

C!P p hq hp x, q p dx q P p d q , (1.7)

i.e. the Poisson process P is the unique solution of the functional equation

C!P P. (1.8)

A further compact formulation, which is equivalent to equation (1.6), is

Px P x . (1.9)

This theorem was firstly given in a general form by Mecke [Mec67], seealso [DVJ08b, prop. 13.1.VII], and leads to the interpretation that thelocal Palm distribution Px of the Poisson process P is the Poisson processwith an additional point at x. Later this characterisation was generalisede.g. by Nguyen, Zessin [NZ79] to Gibbs processes.

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1. Point Processes

1.2.2. Disintegration with respect to the Point Process:Papangelou Kernels

Meckes characterisation of the Poisson process in the version of equa-tion (1.7) or (1.8), respectively, states that the reduced Campbell measureC!P

is absolutely continuous with respect to P, and the Radon-Nikodymderivative is exactly the intensity measure. In general this absolute continu-ity does not hold. But by the definition of the reduced Campbell measurefor B1, B2 B0, C!P p B1 q 3 C!P p B2 q holds whenever B1 B2. Forthe following discussion even the following condition is required:

Definition 1.22(Condition

p

I

q

).

A point processP

is said to satisfy theconditionpI

q, if

p I q C!P p B q 3 P d B B0holds.

ConditionpI

qensures the absolute continuity C!P p B q 3 P for each

BB0 p Xq and therefore the Radon-Nikodym derivative can be com-

puted [Kal78].

Theorem and Definition 1.23 (Papangelou kernel). Let the point processP satisfy p I q , then there exists a measurable mapping

: M p Xq Mp Xq , p , qsuch that

dC!P p B q

dP

p

, Bq

for B

B0 p Xq . Since the paper [MWM79] has been called Papangelou

kernel for the point process P.

By the Radon-Nikodym theorem the Papangelou kernel for P is P-a.s.unique. As in the previous subsection the Palm kernel is interpreted asthe point process conditioned on the occurrence of a point at some site, the

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Papangelou kernel is interpreted as the conditional intensity measure (on X)conditioned on a given configuration. The following theorem from [MWM79]

relates the Papangelou kernel with a partial integration formula for CP.

Theorem 1.24 (Partial Integration). Let P be a point process and be ameasurable mappingM p Xq M p Xq . Then the following statements areequivalent:

i) is a Papangelou kernel for P

ii) P satisfies the partial integration formula for non-negative, measur-

able h

CP p hq

hp x, q p dx q Pp d q

hp x, x q p , dx q Pp dq

By Matthes et al. [MWM79] the last equivalence leads to a nice charac-terisation of simple point processes involving the Papangelou kernel.

Corollary 1.25 (Simplicity). Let be a Papangelou kernel for the point

process P, then the mapping

p

, supp q

is measurable and the sim-plicity of P is equivalent to P{ p , supp q 0} 0.This can be seen by setting h the indicator on pairs

px,

qfor which

p xq 1. For Poisson processes P this corollary implies the known factthat P is a simple point process if and only if the intensity measure ,which is a Papangelou kernel for P, is a diffuse measure.

1.2.3. Construction of Point Processes from Papangelou Kernels

In [MWM79] in general the existence of the point process P, for which thePapangelou kernel is constructed, is assumed. An problem to be addressedin this section is the reverse task firstly developed in [Zes09]: Given a kernel

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1. Point Processes

, construct a point process P, such that is a Papangelou kernel for P.Throughout this subsection let

: M p Xq M p Xqbe a measurable mapping. Let us firstly derive some properties of thefollowing iterated kernels.

Definition 1.26 (Iterated kernel). For a Papangelou kernel : M p Xq M

p Xq and m N let the iterated kernel p mq , m 1, of be

p mqp

, q

: p x1 . . . xm q p x1 . . . xm 1 , dxm q

p x1, dx2 q p , dx1 q .(1.10)

Set p 0q : 1.

The mapping p mq is measurable, and moreover is finite on rectangles ofbounded sets and for m

2 even a symmetric measure [MWM79].

Theorem 1.27. Let be a Papangelou kernel for a point process P. Thenfor every m 1

P

p mq p , B1 . . . Bm q V d B1, . . . , Bm B0 p Xq 1and for m 2

P

p mqp

, B1 . . . Bm q p

mq

p , B

p

1q

. . .

B

p

mq

q

1

for every permutation on{1, . . . , m} and bounded, measurable sets B1, . . .Bm.

In order to construct a point process P for which the measurable map-ping : M p

Xq

Mp

Xq

is a Papangelou kernel, the kernels p mq givenby equation (1.10) at least need to satisfy the properties of the previoustheorem P-a.s. Particularly the symmetry of is assumed for all , whichis equivalent to the cocyle condition,

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Lemma 1.28 (Cocycle condition). Let p mq be given by equation (1.10)for some measurable mapping :

M

p Xq

Mp Xq . Then p mq p , q is a

symmetric measure for each m N, i.e.

p mq

, B1 . . . Bm

p mq

, B p 1 q . . . B p mq

for every permutation on {1, . . . , m} if and only if the cocycle conditionholds

p

x, dy q p , dxq p y, dxq p , dy q .

Proof. Assume firstly that p mq p , q is a symmetric measure for each .

Then by choosing m

2,1B1 p x1 q 1B2 p x2 q p x1 , dx2 q p , dx1 q

p

2q

p , B1 B2 q

p 2q

p, B2 B1 q

1B2 p x1 q 1B1 p x2 q p x1, dx2 q p , dx1 q ,

for all bounded, measurable B1, B2. Hence the cocycle condition holds.Secondly assume that the cocyle condition holds for , then

p

mq

p

, B1

Bmq

1B1 p x1 q 1B2 p x2 q 1Bm p xm q

p x1 . . . xm 1, dx2 q p x1 , dx2 q p , dx1 q

1B1 p x1 q 1B2 p x2 q 1Bm p xm q

p x1 . . . xm 1, dx2 q p x2 , dx1 q p , dx2 q

p mq

p, B2 B1 Bm q

due to the cocycle condition. This equation holds for every next neighbortransposition and hence by iteration for all transpositions. Since everypermutation has a decomposition into transpositions, the equation holdsfor all permutations, therefore p mq p , q is a symmetric measure.

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1. Point Processes

In the previous subsection the Papangelou kernel was obtained as thedisintegration of the reduced Campbell measure permitting the interpre-

tation that p , q is the conditional intensity measure conditioned on theconfiguration . Thus in the following plays the role of a boundary con-dition, for which the point process is constructed.

Let M p Xq and Zp mq p q be the mass of the iterated kernel p mq aswell as p q the possibly infinite limit of the series,

Zp mqp

q

:

p m qp

, X

. . .

Xq

, p

q

:

m

0

Zp mq p q

m!. (1.11)

is called integrable if p

q V

for each M

p

Xq

, and in this case thepoint process P given by

P p q :1

p q

m

0

1

m!

Xm

p x1 . . . xm q p

mq

, dx1, . . . , dxm

(1.12)

is well-defined. By [Zes09]:

Proposition 1.29. Let : M p Xq Mf be a finite kernel satisfying thecocycle condition and assume to be integrable. Then for every boundary

configuration

M

p Xq,

P

is a solution of the partial integration formula

CP p hq

hp x, x q p , dxq Pp dq .

Definition 1.30 (Papangelou process for kernel ). P is the (finite) Pa-pangelou process for the symmetric and integrable kernel .

Proof of proposition 1.29. By the definition of the Campbell measure

CP p hq hp x, q p dxq P p dq

1

p q

m

0

1

m!

Xm

mj

1

hp xj , x1 . . . xm q

p mq

, dx1, . . . , dxm

,

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then firstly by the symmetry of p mq and secondly by integration and thedefinition of the iterated kernels

1

p q

m 0

1

pm

1q !

Xm

hp

xm, x1 . . . xm q

p m q

, dx1, . . . , dxm

1

p

q

m

0

1

pm

1

q!

Xm 1

X

hp xm, x1 . . . xm q

p

x1 . . . xm 1, dxm q p

m

1q

, dx1, . . . , dxm

hp x, x q p , dxq P p dq .The aim is to extend the construction of finite Papangelou processes for

finite kernels to kernels which are -finite. The strategy is to constructthe process locally and then to glue these locally defined processes together.For a bounded and measurable B let the -algebra of the events inside Bbe

EB

B I : B

I

B, B I

B0 p Xq

and define the restriction to B for

EB-measurable as

B p , q :m

0

1

m!

Bm

p

x1 . . . xm q p

mq

Bc , dx1, . . . , dxm

. (1.13)

Note that the mass of in B is cut. The definitions of the normalisation

constants Zp

mq

B p q and B p q in equation (1.11) carry over directly. iscalled locally integrable if for each bounded B and M p Xq , B p q isfinite. A construction similar to the Poisson process can be used in caseof additional measurability conditions on B, that is if B cannot see what

happens outside B.

Theorem 1.31 (Papangelou processes with independent increments). Letthe measurable mapping : M p Xq Mp Xq be locally integrable andsatisfy the cocycle condition. If in addition B defined in equation (1.13)

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1. Point Processes

is

EB-measurable, then there exists a point process P on X which is in-

dependent of the boundary configuration

M

p Xq and for which is a

Papangelou kernel, i.e. P satisfies the partial integration formula,

CP p hq

hp x, x q p , dx q Pp dq .

Proof. Two main steps have to be done: Firstly P has to be constructed andsecondly the partial integration formula for P has to be shown. For the firstpart let p Bn q n

0 be a locally finite partition of X of bounded sets, i.e. foreach bounded set B

B0 p Xq , B Bn $ r only for finitely many n. Then

on each Bn a point process P

n is constructed according to equation (1.13),

Pn p q :1

Bn p qBn p , q .

Due to the measurability condition on B,Bmn

p

x1 . . . xm q p

mq

Bcn , dx1, . . . , dxm

Bmn p x1 . . . xm q p mq 0, dx1, . . . , dxm

is independent of and the superscript of Pn may be dropped.

Thus for each n a point process Pn on M p Bn q is constructed. Let

N:

n

0

M p

Bn q

the product space. By the Daniell-Kolmogorov extension theorem, there ex-

ists a probability measure P on the product space Nwith finite-dimensionaldistributions given by the corresponding product of the Pns. Finally mapP via

N

M p

Xq

,p

n q n

1

n

0

n

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to obtain a probability measure on M p Xq , which also will be denoted byP with abuse of notation. It remains to show that is the Papangelou

kernel for P.First of all, let h be of the form h

px,

q g

px

q

p

qwith supp g

Bj for

some j 1 and being EB0 ... Bm-measurable for some m j. ThenCPp hq

n

0

h

x,

Vk 0

k

k p dx q Pp dq

Bjg

px

q

V

k 0k

j p dxq Pp d q

since supp g Bj and furthermore because of being EB1 ... Bm-measur-able,

Bj

gp

xq

mk

0

k

j p dxq Pp d q ,

for which P can now be replaced by P1 Pm. Finally the application

of partial integration for Pj yields

Bj

gp

xq

mk

0

k

j p dxq P0 p d0 q Pm p dm q

Bj

gp

xq

mk

0

k x

pj , dx q P0 p d0 q Pm p dm q

Bj g p xq m

k 0 k xm

k 0 k, dxP0 p d0 q Pm p dm q

g p xq p q p , dx q Pp dq .

This result extends to general non-negative and measurable h.

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1. Point Processes

Particularly in the step of the application of the partial integration for-mula the measurability condition on B, and hence the independence sim-

plified a lot. For general , that is B not necessarily EB-measurable, finerinstruments are necessary. In fact, proposition 1.31 is a special case ofproposition 1.33. In [Zes09] the theorem is only given for the boundaryconfiguration 0, here we drop this restriction. Furthermore an addi-tional assumption on the normalisation constants seems to be required incontrast to [Zes09]. The main schedule, firstly to construct the global pro-cess and secondly to show the partial integration formula, stays the same.The means of theorem 1.31 have to be refined: a Markov construction to-gether with the theorem of Ionescu Tulcea [Kal02, thm. 6.17] yields thefirst part, for the second further assumptions are necessary.

Theorem 1.32 (Ionescu Tulcea). For any measurable spaces p Sn, Sn q andprobability kernels n from S1 Sn 1 to Sn, n N, there exist some

random elements n in Sn, n N, such that p 1, . . . , n qd

1 n for

all n.

A kernel is said to satisfy the Feller condition if for every increasingsequence p Bn q n of bounded sets which exhausts X,

p Bn , q p , q

vaguely as n V

.A further condition needs to be discussed: If p y, q is absolutely

continuous with respect to p , q outside {y} for every boundary configu-ration M p Xq ,

1{y}c p xq p y, dxq 1{y}c p xq f p y, xq p , dx q , (1.14)

then for bounded, measurable B and x

B,

B p x, q m

0

1

m!

Bm

p x1 . . . xm Bc x q

p mq

Bc x, dx1, . . . , dxm

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m 01

m! Bm

px1 . . . xm Bc x q

f p x, xm q f p x, x1 q p

mq

Bc , dx1, . . . , dxm

B

, p x q f p x, q

,

where f p x, x1 . . . xm q : f p x, x1 q f p x, xm q . Therefore, if PB

:

B p q

1B p , q ,

P xB p q

B p q

B p x qPB

p x q f p x, q

. (1.15)

Theorem 1.33 (General Papangelou processes). Assume that the measur-able mapping : M p Xq M p Xq satisfies the cocycle condition and theFeller condition, and is locally integrable. Let M p Xq be a given bound-ary configuration. If furthermore p y, q is absolutely continuous withrespect to p , q outside{y} as in equation (1.14), and for eachB B0 p Xq ,x

Bc the normalisation constants satisfy B p q B p x q , then there

exists a point process P on X for which is a Papangelou kernel, i.e. P

satisfies the partial integration formula

CPp

hq hp x, x q p , dx q Pp dq .

Proof. As in the proof of proposition 1.31 let p Bn q n

0 be a locally finitepartition of X of bounded sets. Then the following finite point processesexist by equation (1.12)

Pn p q 1

n p q

m

0

1

m!

Bmn

p x1 . . . nm q p mq

Bcn , dx1, . . . , dxm

.

(1.16)

Successively on each Bn a finite point process will be constructed with theboundary condition in the regions B0, . . . , Bn 1 replaced by a realisa-tion of the corresponding, already constructed, finite point processes onB0, . . . , Bn

1. LetQ0 p d0 q : P

0

p d0 q ,

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1. Point Processes

the dependence on suppressed for the moment. Furthermore denote bym the sum 0 . . . m and by

p mq the restriction of to the complement

of B0 . . . Bm. For p 0, . . . , m

1 q M p B0 q M p Bm

1 q let

Qm p 0, . . . , m

1; dm q : Pp m 1q

m 1

m p dm q , (1.17)

i.e. Qm is a probability kernel M p B0 q M p Bm

1 q M p Bm q .Since by equation (1.16) any contribution of inside Bm is cut, in thedefinition (1.17) p m 1 q may be replaced by p mq . The choice p m 1q isconsistent with the definition of Q0.

By the theorem of Ionescu Tulcea there exists a probability measure Pon N such that its finite-dimensional distributions P

p

0,...,nq

are given by

P

p

0,...,nq

p d0, . . . , dn q Pp n 1q n 1

n p dn q Pp 0q

1p d1 q P

0

p d0 q .

For simplicity P again is identified with its image under the mapping

N M p Xq , p n q n

0 n 0n.

Therefore the point process P exists, considered either as a point processon Nor M p Xq , respectively. The partial integration formula remains tobe shown.

In choosing again h to be of the form hp x, q g p xq p q with supp g Bjfor some j 1 and being EB0 ... Bm-measurable for some m j, the firstlines of the proof of the partial integration in the proof of proposition 1.31can be followed, and then continued by

CP p hq

Bj

gp

xq

m

j p dx q Pp m 1q

m 1

m p dm q

Pp 0q

1p d1 q P

0

p d0 q ,

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for which the partial integration formula for finite Papangelou processescan be applied for the j-th kernel

Bj

g p x q

m x

Pp m 1q

m 1 xm p dm q

Pp j q j xj 1 p dj 1 q p

j , dxq

Pp j 1q j 1

j p dj q

Pp 0 q

1p d1 q P

0

p d0 q .

Because of x

Bj 1, by equation (1.15) and the equality of the normal-isation constants, and the integrations can be exchanged such that firstly

with respect to x is integrated,

Bj

g p x q

m x mk j 1

f p x, k q p j , dxq

Pp

m 1q

m 1

m p dn q Pp 0q

1p d1 q P

0

p d0 q

g

px

q

p

q

p, dx

qP

pd

q.

For the last line observe that by assumption

f p x, j 1 q p j , dx

q

pj 1, dx

q.

Indeed, the EB-measurability in proposition 1.31 implies Feller and ab-solute continuity condition, and therefore proposition 1.31 is a special caseof proposition 1.33.

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2. Deviation Principles

One of the main techniques used in the later parts is the principle of large de-viations. Basically, the situation is the following: given a sequence

pYk q k 1

of identically distributed, uncorrelated random variables with finite secondmoment, the weak law of large numbers states that the average of the firstn variables 1nSn tends to the expectation EY1 weakly as n V . Clearlythe probability that the average 1

nSn stays away from EY1, tends to 0. Such

events {|

1

nSn EY1 | } are called rare events, untypical events or largedeviations. The basic question is: What is the probability of a rare eventand at which speed does it vanish?

Cramers theorem (see e.g. Dembo and Zeitouni [DZ98, thm. 2.2.3]or Deuschel and Stroock [DS00, thm. 1.2.6]) states that this probabilitybehaves like exp

n inf{Ip

xq

:|x

EY1 | }

, where I is a non-negative

function called rate function. Several observations can be made: Firstly, theprobability of this rare event decays exponentially fast, i.e. 1nSn may deviatewith only exponentially small probability. Secondly, inf{Ip x q : x R} 0and moreover, the infimum is in fact a minimum. The limit of the 1nSnsoccurs as the minimum of I, which is at the same time a zero of I. Finallythe theorem is not restricted to only random variables; for random vectorsCramers theorem is also valid and allows extensions to projective limits(which are not trivial indeed).

Particularly the second observation, the determination of a weak limit

as the minimiser of an optimisation problem, will be important: the weaklimit is exactly the minimiser of I. Conversely, determining the minimiserof I means to find a weak limit. Moreover, if a condition on the 1nSnis present, the large deviation principle leads to an optimisation problemwith constraints.

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In sections 4.5, 7.2 and 7.3, large deviation principles are a basic tool forthe derivation of limit theorems. The large deviations for Poisson processes

at high intensity, see [GW95], and required adaptations are discussed insection 2.2. For the discussion of the asymptotic behaviour of the model tobe introduced in chapter 3 large and small deviations for Brownian bridgesare given in subsection 2.3.

2.1. General Large Deviation Principles

In the situation of the uncorrelated, identically distributed random vari-ables, the rate function I can be shown to satisfy two important properties:convexity and the compactness of the level sets {I u}. This ensures theexistence of a minimiser, which is, in the situation above, unique and at thesame time the (unique) zero ofI. To obtain suitable optimisation problemsand for I to encode the behaviour of a family of probability measures, thefollowing properties are required:

Definition 2.1 (Rate function). A rate functionI is a lower semicontinuousmapping I : Y 0, V s , i.e. the level sets {y Y : Ip y q } are closedfor every non-negative . I is furthermore a good rate function, if the levelsets are compact.

Over closed sets good rate functions achieve its minimum, which impliesthat a weak law of large numbers holds. Assume p Pn q n being a sequenceof probability measures on

Y, B p Y q

with B p Y q complete. The precise

definition of the large deviation principle is the following

Definition 2.2 (Large Deviation Principle). The sequencep

Pn q n of proba-bility measures satisfies a large deviation principle with good rate functionI and speed an if the following two bounds hold for every G B p Y q open

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2.1. General Large Deviation Principles

and every F B p Y q closed:

lim infn V

1

|an |

log PnG infy

G Ip y q (2.1)

lim supn

V

1

| an |log Pn

F

infy

FIp y q . (2.2)

By the lower semicontinuity, see e.g. Deuschel and Stroock [DS00, lemma2.1.1], the rate function can be shown to be unique in case of existence oncethe speed is fixed. Therefore I is said to govern the large deviations ofp

Pn q n.The main job is to determine the rate function I. Consider Pn to be the

law ofSn as in the example at the beginning, then by the Markov inequality

P

Sn nx

P

euSn eunx

e unx E euSn exp

n

ux logp uq

,

where p u q : log E euY1 is the logarithmic moment generating function.Optimising the exponent on the rhs. with respect to u yields a candidatefor the rate function I governing the large deviations of p Sn q n. Indeed,basically Cramers theorem states that the convex conjugate of is therate function,

Ip

xq

p

xq

:

sup{ux

logp

uq

: u R

}.and the speed can be chosen to be an n. Furthermore I can be shownto have exactly one minimiser given by EY1, which is simultaneously theunique zero. Roughly speaking, Sn satisfying a large deviation principlewith rate function I means

P

Sn A

e n infx A Ip xq .

Such a statement remains true in greater generality for measures on aspace Y. Let Y be the dual ofY, then for a probability measure P let P

be its logarithmic moment generating function

P : Y

R P p uq : log

Y

expu, yPp dy q ,

where u, y : up y q .

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Definition 2.3 (Fenchel-Legendre transform). For a sequence p Pn q n ofprobability measures and an increasing sequence p an q n of positive real num-

bers let

p

uq

:

limn

V

1

anPn p u{ an q

The Fenchel-Legendre transform of is the convex conjugate of ,

p

yq

:

sup{u, y

p

uq

: u

Y }.

In the following we need to assume that is well-defined and finite inan open set containing 0, Gateaux-differentiable and lower semicontinuous.The rate function then is exactly , see e.g. [DZ98, thm. 4.5.27].

Theorem 2.4 (Gartner-Ellis). Let p Pn q n be an exponentially tight sequenceof probability measures. If exists in a neighborhood of 0, is Gateaux-differentiable and lower semicontinuous, then p Pn q n satisfies a large devia-tion principle with good rate function .

2.2. Large Deviations for Poisson Processes at

increasing Intensity

In the particular situations in sections 4.5, 7.2 and 7.3 Poisson processeswith increasing intensity are given. Assume Pr being a Poisson process withintensity measure r, r 0. As r V , the expected number of particlesB in a bounded region B grows by the same factor r and

Br p B q by

the law of large numbers. Therefore

Pr :

PrBr is a candidate for a large deviation principle. The result of Guo andWu [GW95] even states

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2.2. Large Deviations for Poisson Processes at increasing Intensity

Theorem 2.5 (Large deviation principle for Poisson processes at high in-

tensity). As r V , Prr satisfies a large deviation principle onMp

Xq with speed r and rate function Ip ; q : M p Xq 0, V s ,

Ip

; q

pf log f

f

1

q if 3 , f : dd, f log f f 1 L

1p

q

V otherwise.

The function Ip ; q is called relative entropy with respect to and agreeswith . Because of some necessary comments on that result, the mainpoints of the proof are demonstrated

Sketch of Proof. The moment generating function for the Poisson processwas given in section 1.1.3, and therefore passing to the limit yields

p fq

X

1 ef

d (2.3)

for any continuous f with bounded support. Moreover, is Gateaux-differentiable in a neighborhood of f with

dp fq g s :d

dt

p f tg q | t

0 Xg efd.Therefore Pr

r

satisfies a large deviation principle with rate func-

tion , which can be identified as the relative entropy with respect to by solving the variational principle.

Remark 2.6. Instead of the intensity measure r for an increasing factor r asequence p n q n of intensity measures with

nn is sufficient for the limit

in equation (2.3) to exist, which is exactly the situation in section 4.5.

Particularly the case X

N is important for sections 4.5, 7.2 and 7.3,where the intensity measure is a finite measure of the form

j

1

zj

jj (2.4)

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for certain parameters z p 0, 1s and 0. There even stronger resultshold and are required: In section 4.5, z 1 and 2, the test functions are

allowed to have an unbounded support, but stay bounded; and in chapter 7,

1 and z

1, the test functions are allowed to grow linearly. These

stronger results have to be justified.In the first case, p j q j for some 2, for f and g bounded the

following two estimates hold

p fq j

1

j

efp j q 1

ec1 p Nq V

dp

fq

gs j 1j g p j q efp j q c2 ec1 p Nq V

for constants c1 | f| and c2 | g | . The first estimate ensures that thedomain of contains an open neighborhood of 0, the second ensures thedifferentiability.

Corollary 2.7 (Large Deviations for weak topology). The large deviationprinciple of the family of Poisson processes Pr, r 0, onN with givenby equation (2.4) for 2 and z 1 in theorem 2.5 holds true onM p Nqequipped with the topology of weak convergence.

For the second case, p

jq

zj

j , let | fp j q | c1 p 1 j q and | g p j q | c2 p 1 j q ,then

p fq j

1

zj

j

efp j q 1

j

1

zj

jec1 p 1 j q

dp fq g s j

1

zj

jg p j q efp j q c2

j

1

zj ec1 p 1 j q .

For sufficiently small c1, the rhs. of the first equation converges, hence thedomain of contains an open neighborhood of 0, and the second estimateyields the differentiability of in the domain of its convergence. Let the -topology be the topology on Mp Nq generated by these at most linearlygrowning functions.

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2.3. Deviations for Brownian Motions and Brownian Bridges

Corollary 2.8 (Large Deviations for -topology). The large deviationprinciple of the family of Poisson processes Pr, r 0, onN with given

by equation (2.4) for 1 and z 1 in theorem 2.5 holds true onM p Nqequipped with the topology of -convergence.

2.3. Deviations for Brownian Motions and Brownian

Bridges

The model to be introduced in section 3.1 deals with measures on Brownianloop spaces. Particular large deviation principles for Brownian motion can

serve information about the asymptotic behaviour of these measures forincreasing and decreasing loop lengths, respectively. The behaviour forlong loops relies on theorem 2.11, which is a generalisation of Schilderstheorem, see e.g. Dembo and Zeitouni [DZ98, thm. 5.2.3].

Theorem 2.9 (Large deviations of Brownian motion, Schilder). Let denoteby p Wt q t r 0,1 s a Brownian motion. Then for Borel measurable A,

lim inf

02 logPp W B q inf

y

intAIp y q ,

lim sup

0

2 logPp W B q infy

clAI

py

q,

where Ip y q

Wy22

is a good rate function.

Schilders theorem is a particular case of a large deviation principle forgeneral centered Gaussian measures on separable, real Banach spaces [DS00,thm. 3.4.12]

Definition 2.10 (Wiener quadruple).p

E , H , S , P q

is a Wiener quadrupleif

i) E is a separable, real Banach space,

ii) H is a separable, real Hilbert space,

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iii) S : H E is continuous, linear and injective,

iv) P is a Gaussian measure on R, i.e.E

exp[i, x] Pp dxq exp

1

2S 2H

for all

E , where S : E

H is the adjoint map of S.

Theorem 2.11 (Large Deviations of centered Gaussian processes). If P isa centered Gaussian measure on the separable, real Banach space E, thenthere exist a separable, real Hilbert space H and a continuous, linear injec-

tion S : H

E such thatp

E , H , S , P q

is a Wiener quadruple. Moreover, ifp

E , H , S , P q is any Wiener quadruple, then S is a compact map, satisfies

S

E

x2Ep dxq1

{

2

and the family p p q q n 1 satisfies a large deviation principle with rate func-tion

p

xq

1

2S 1x2H if x Sp Hq

V if x E

\S

pH

q

For the the Wiener measure WT on 0, Ts , E {x C 0, Ts ,Rd :xp 0q 0}, H is the space of absolutely continuous functions whose deriva-tives L2-norm is bounded and S is given by

Sh

p tq

t0

hp sq ds.

S is computed to be

S p tq T

t p

dsq

,

from which the covariance follows as

Qp

, Iq

:S , S I

T0

T0

s

t Ipds

q

pdt

q. (2.5)

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2.3. Deviations for Brownian Motions and Brownian Bridges

In case of the Brownian bridge on 0, Ts , E gets the additional conditionxp Tq 0, H gets the condition that the integral over 0, Ts vanishes, and

the kernel for the covariance in equation (2.5) is replaced by s t stT .Particularly S 1 is still the derivative.

Therefore, if B p

Bt q t r

01,s

is a Brownian bridge, then the probability ofthe event {supt

r

0,1s

| Bt | L} vanishes exponentially fast as L V . Moreprecisely

limL V

1

L2logP

supt

r

0,1s

| Bt | L

2,

and Brownian motions and bridges are very unlikely to leave a large region

at least once. On the other hand both processes are very likely to leave verysmall regions. The precise statements about the small ball probabilities aregiven by Li and Shao [LS01, thm. 6.3].

Theorem 2.12 (Small deviations of Brownian bridges; Shao, Li). Let de-note by

Gaussian Loop- and Pólya Processes- A Point Process Approach

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