Donsker’s Invariance Principle and Brownian martingales
Transcript of Donsker’s Invariance Principle and Brownian martingales
Donsker’s Invariance Principleand Brownian martingales
Updated June 2, 2021
Plan 0
Donsker’s Invariance PrincipleWeak convergence in Wiener spaceTools for verifying tightnessContinuous-time martingalesExamples using Brownian motion
Scaling limit of random walks 1
Brownian motion constructed as a Cpr0, 8qq-valued r.v.Original motivation: scaling limit of random walksLet Z1, Z2, . . . be i.i.d. R-valued r.v.’s and set
@n P N : Xn :“nÿ
k“1
Zk
Use these to construct an element of Cpr0, 8qq via
@t P r0, 8q : Ypnqt :“ 1?
n
´Xtntu ` pnt ´ tntuqXtntu`1
¯
Consequences of CLT 2
If EpZ1q “ 0 and EpZ21q “ s2, then by CLT:
@t • 0 : Ypnqt
law›ÑnÑ8 N p0, s2tq
Note: sBtlaw“ N p0, s2tq.
Mutlivariate CLT even gives convergence in the sense of finitedimensional distributions:
@0 § t1 † ¨ ¨ ¨ † tk : pYpnqt1
, . . . , Ypnqtk
˘ law›ÑnÑ8 psBt1 , . . . , sBtkq
where B is the SBM.
Q: Convergence of the law of t fiÑ Ypnqt on Cpr0, 8qq?
Donsker’s Invariance Principle 3
Theorem (Donsker 1951)For Xn :“ Z1 ` ¨ ¨ ¨ ` Zn with tZkuk•1 i.i.d. satisfying EpZ1q “ 0and EpZ2
1q “ 1, as n Ñ 8 the law of
Ypnqt :“ 1?
n
´Xtntu ` pnt ´ tntuqXtntu`1
¯
on pCpr0, 8qq,BpCpr0, 8qqqq converges weakly to Wiener measure.
Precise meaning of convergence? Define
@A P BpCpr0, 8qqq : PpnqpAq :“ PpYpnq P Aq
Donsker’s Theorem says: Ppnq wÑ PW as n Ñ 8.
Added value 4
Q: Why is this more than just conv. of finite-dim. distributions?
CorollaryFor above setting,
1?n
max1§k§n
Xklaw݄
nÑ8 max0§t§1
Bt
Corollary
Given a ° 0, set Tpnqa :“ inf
k • 0 : Xk • a
(. Then
1n
Tpnqa?
nlaw݄
nÑ8 inftt • 0 : Bt • au
Proof of Corollaries 5
Proof of Corollaries 6
How to prove Donsker’s theorem? 7
Theorem
Let tPpnqun•1 and P be probability measures on the Wiener spacepCpr0, 8qq,BpCpr0, 8qqqq. Then
Ppnq w›ÑnÑ8 P
is equivalent to the conjunction of(1) Ppnq Ñ P in the sense of finite-dimensional distributions(2) tPpnqun•1 is tight
Recall: tPpnqun•1 on pX ,BpX qq is tight if
@e ° 0 DK Ñ X compact : lim supnÑ8
PpnqpX r Kq † e
Proof of Theorem 8
Verifying tightness 9
Arzela-Ascoli Theorem: A set K Ñ Cpr0, 8qq is compact if andonly if for each M • 1 the set
w|r0,Ms : w P K
(
is closed in Cpr0, Msq, pointwise bounded and equicontinuous.
Equicontinuity hard: needs truncation (to increase availablemoments) & Kolmogorov inequality (to control oscillation ofpaths over intervals)
We will prove Donsker’s Theorem via Martingale FunctionalCentral Limit Theorem (to be discussed next time)
Continuous-time martingales 10
DefinitionAn R-valued process tXtut•0 is a martingale with respect tofiltration tFtut•0 if(1) @t • 0 : Xt is Ft-measurable with Xt P L1, and(2) @t, s • 0 : EpXt`s|Ftq “ Xt a.s.
Note:submartingale if EpXt`s|Ftq • Xt, supermartingale if “§”continuous/cadlag (sub/super)martingale if every samplepath t fiÑ Xt is continuous/cadlag
Fact (275D): regularity assumptions sometimes superfluous;sub/supermartingales admit cadlag versions
Brownian martingales 11
The following are continuous martingales
Bt, B2t ´ t, B3
t ´ 3tBt, B4t ´ 6tB2
t ` 3t2, . . .
These are all generated by continuous martingale
Mt :“ elBt´ l22 t
Indeed,
elBt´ l22 t “ 1 ` lBt ` l2
2pB2
t ´ tq
` l3
6pB3
t ´ 3tBtq ` l4
24pB4
t ´ 6tB2t ` 3t2q ` . . .
Basic facts derived via discrete-time martingales 12
LemmaX martingale ñ tXt : t § au UI for all a P p0, 8q
LemmaX cadlag martingale ^ T stopping time ñ tXT^tut•0 martingale
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TO BE CONTINUED . . .