· * Norman Kaplan

A NOTE ON THE DISTRIBUTION OF GENERATIONS

IN A BRANCHING PROCESS

Preprint 1974 No. 13

INSTITUTE OF MATHEMATICAL STATISTICS

UNIVERSITY OF COPENHAGEN

October 1974

* Department of Statistics, University of California,

Berkeley.

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ABSTRACT

Some limit theorems relating to the number of partials in the

a th generation alive at time t are reexamined. A single me­

thod of proof is given and shown to work equally as well for

certain generalizations.

Key words: Age-dependent branching process, generation number.

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Written while on leave at the Institute of Mathematical

Statistics, University of Copenhagen. Research partially

supported by N.S.F. Grant, 31091 X.

SECTION 1

1. Introduction

Let {X(t)}t>O be a supercritical Bellman-Harris branching pro-

cess with age distribution F, and offspring p.g.f. f(s). Some

attention has been given to the study of the limit behavior of

the following random quantities:

and

th Uk(t) = number of particles of the k generation born

before t

number of particles of the kth generation alive

at time t.

Results can be found in [1], [21, [4], [7], [8], [9]. In each

of these papers the technitue of proof is highly analytic. It

turns out that some of the existing theorems can be gotten very

quickly as a consequence of the structure of the process and

the Berry-E:s'S',eertT,heorem. This technLque we feel is the natural

one since it is simple and applies as well to a generalization

of the process where the lifetime distribution of each genera­

tion is varible. This case was recently studied by Fildes [4].

We now introduce the necessary notation and state our result for

the clAssical case. Put:

00

f i ll. v dF(v),

1

(j2 =

<p (x)

0

ll2

1

/27f

2 - (lll) ,

1 2 --x e 2

m. tJ"i f (1-)

1 als

m = ml

x and <l>(x) = f <p(v)dv,-oo < x < 00

-00

-3-

If Xk = number of particles in the kth generation, then it is -k

well known [6] that lim m Xk = w.p.1 and if m2 < 00, the limit k~

exists in mean square.

Theorem. Assume~. < --- ~ 00, m.

~ < 00 i 1,2, m1 > 1 and m3 < 00.

Let tk = k~l + crlkt. Th~,

(1. 1)

Suppose in addition F is non lattice. Then,

(1. 2)

SECTION 2

Proof of the Theorem L2

We first prove (1.1). -k

Since m Xk -+ W, it suffices to show,

Following the notation of [6, Chapter 5], it ~s not difficult

to check that, n.

no~l

2: 2: i 1 =1 i 2 =1

where

T.. . is the length of life of <!:k> ~1~2···~k-1 ~

n.. . is the number of offspring produced by <!:k> ~1~2"'~k-1 --

IA is the indicator function of the set A.

We introduce some notation. Put for k > 1,

and

Then,

n o L

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n.. . ~t~2·· ·~k-2

L

ik_l=l

For any collection of random variables G, let a(G) be the

a-field generated by G. Define for k > 1,

{en. ,T~ ) for all <i.> such that j < k}. ~. :to -J ... J .... J

Then,

Conditioned on Gk - l , Uk(t k ) is just the sum of independent ran­

dom variables with the typical term having variance equal to

Hence,

where * represents convolution and F(k)(t)

k > 1. Also observe that

F * F(k_l)(t),

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The last expression 1S of exactly the same form as C2.I) except

that the summation is over one less index. Thus we can repeat

the calculation k-I times to obtain

e = E(IUCt ) - ~(t)Xk]2) = ~ mk-jE(X~)B. (k) k . k j=I J J

where

B. (k) J

2 2 F(j_l) * F(k-j+I)(t k ) - F(j) * F(k_j)(t k )

and 1 < j < k-I

I

\

2 2 . It is easy to show that E(X .) = Oem J). Therefore, there

J exists some constant A independent ofk such that

m- 2ke < A k

k L

j=I m-(k-j) B.(k)

J

Let E > O. Choose N o

such that L m- J < E

Then for k > N , o

m- 2ke < 2EA + k

j > N o

k L m(k-j)B.(k)

j =k-N+I J o

(2.2)

We now apply the Ber:ry Ess€en\Theorem[5, pg. 201] to conclude

that

lim B. (k) k~oo J

This proves (1.1).

0, k - N +1 < j < k o

";;6-

To prove (1.2) we first observe that

IkOVk(t k ) - mcp(t)Xk = ~ (IkOI{S. <t k < S. } -m¢(t))

~k+l ~k ~'+l

Thus if we condition exactly as before we obtain,

where

and

= k L

j=O

Let N > O. Then using the extended :B~rry Es'Seen> !fheo·rem o

[5, pg 210] it is not difficult to show that

lim C. (k) k+oo J

and

sup IC.(k) I ~ Dl ~ l<j<k-N J J

- - 0

where Dl is a constant independent of k.

Hence,

k-N -2k (0 m-(k-j) _k

J.) lim sup(m Ak ) = lim supfD 2 L

k+oo k+oo j = 1

where D2 is a suitable constant. Let E > 0 such that

(2.3)

and

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-1 Then for k > [E ] ,

[k~E-1]-1. -N "I: m-] k -<m 0 k 1

k-]' k-N 1-r j=N 0 o

k-l I:

j=[k-E- 1 J-l

-1 m- j k -< D k .-[k-£ J

k-j 3

for D3 a suitable constant. Since N is arbitrary we are done. o

SEG'lION 3

A Generalization.

Suppose a particle in generation n lives a random length of ti­

me governed by F which depends on n. One can easily check n

that expressions similar to (2.2) and (2~3) hold in this more "

general setup. In fact, the only thing that onet~eeds to carry

through the arguments oft thee previ~tts s~eti~n -is -a,Bce;;r:r~Y5'~~~:€'e1't "4:0''­

pe ,theorem for non-identically distributed random variables.

Fortunately such results exist under suitable assumptions.

[3, pgg 78 and 84].

We now intvoduce some notation and state the result. Let

].1. 1.

f v dF. (v) 1.

k I: ].1.

i=1 1.

t = m + k k

2 cr. 1.

fv 2 () 2 dF. v -].1 •• 1. 1.

k 2 I: cr.

i=1 1.

Theorem. Assume there exist constants 0 < A < B < 00 such that

A < inf 2

and f 3 dF. (v) B . cr. sup v <

1. i 1. 1.

Then

lim rUk(t k )

W<l?(t)]2) 0 E\l k - = k-+oo m.

Assume in addition

-8-

sup J v 4 dF. (v) t oo,F ~ 1 n w

1

F,limsuplf(w)l<l Iwl+oo

and for some L,

lim'{sup Ifk(W) - f(w)1} = 0 k+oo I WI >L

where

Je ivw = dFk(v), few) = Je ivW dF(v).

( S V (t ) )

lim E [ k: k - ~W ~(t)]2 k+oo m

o

where ~ = Jv dF(v).

Proof. The conditions of the theorem are sufficient to imply the

required versions of the };j',erry Ks:se'g;flTheor;e;mgiven in (3). Q.E.D.

Remark. The assumptions of Fildes imply ours.

Conjecture. I believe that this method can be applied to the

generalized age-dependent model to obtain analogous results.

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BIBLIOGRAPHY

[1] Buhler, W.J. (1970). The distribution of generations and

other aspects of the family structure of a th branching process. Proc. of 6 Berkely Sy.

Math.Stat.Prob. 3, 463-480.

[2] Buhler, W.J. (1971). Generations and degree of relation-

ship in a supercriticalMarkov branching pro­

cess. Z.Wahr.Verw.Geb. 8, 141-152.

[ 3 ] {i!t aile'Jr j Ii.. ('1',$'11 i,lf~d'~~~'f+~i'1l.1'it~~~ 'a,'!:l4'jit:~~'i)l1}~,t:w ~j ~ , , .• b"~' , '3~d " ':l' 'OJ ;;. .' •. • " • e!t17 l:"~~'~'O:~o: ....," Ed lC t''It('Ul'"",{l allb t7#i: d g e i {1n 1: ve'1i'''';;,

~11:' 1~:Py; e ~s: ,;~{J;a.mlJ r i Ii g e, Eng 1 a 111 d.

[IJ Fildes, R. (1974). An age-dependent branching process

with varible lifetime distribution: the gene­

ration size. Adv. in App. Probe 291-308.

[5] Gnedenko B.V. and Kolmogorov A.N. (1954). Limit Distribu-

tions for Sums of Independent Random Variables.

Addison Wesley, Reading Mass.

[6] Harris, T.E. (1963). The TheorE of Branching Processes.

Springer Verlag, Berlin.

[7] Kharlamov, B.P. (1969). On the generation numbers of par­

ticles in a branching process with overlapping

generations. Theor.Prob.Appl. 14, 44-50.

[8] Ma r tin - L 0 f, A. (1966). A limit theorem for the size of

the nth generation of an age-dependent bran­

ching process. J.Math.Anal.Appl. 12,273-279.

[9] Samuels, M. L. (1971). Distribution of the branching pro­

cess population among generations. J.Appl.

Probe 8, 655-667.