Equilibrium Distributions of Age Dependent Galton Watson Processes, II

Math. Nachr. 160 (1993) 313-324

Equilibrium Distributions of Age Dependent GALTON WATSON Processes, I1

By KLAUS MATTHES and RAINER SIEGMUND-SCHULTZE of Berlin and ANTON WAKOLBINGER of Linz

(Received March 4, 1992)

Introduction

In the present paper two questions concerning equilibrium distributions P will find an answer, which occurred implicitly already in the first part [6]. In both cases the intuitively expected answer is easy to get under the additional assumption of P being a first order distribution, i.e. the finiteness of the first moment measure at each site of the phase space. But the results of [6] show that “pathological” effects are possible in the absence of this condition.

In section 9. for an arbitrary discrete model (cf. section 1.) we introduced an equi- valence relation in the phase space A by means of the given stochastic branching dy- namics K. The definition was the following: For two sites a, b in A we say that b can bereachedfrorn a ifthere is a number n EZ+ such that the n-th generation ofdescendants of an a-individual (of an individual at site a) occupies site b with a positive probability. We call a and b equivalent, if each of these sites can be reached from the other one. Since the relation of reachability can be expressed in terms of the intensity kernel J of K, we speak of J-equivalence in A .

Consider an arbitrary equilibrium distribution P and any U E A with P ( @ ( { a } ) > O ) > 0 (i.e. a ~ s u p p A ~ ) . By proposition 9.1. we may conclude that the J-equivalence class A,,, of a cannot be reached from any b E (supp AP)\A[,,, supposed that the embedded y[,l-equilibrium P,,, is hermetic (see sections 3. and 4.). This means that the hermeticity of the embedded equilibrium implies that A,,, is P-autonomous. In [6] the question was not decided, whether these assumptions imply that Plbl is hermetic for all b E A,,,, i.e. whether hermeticity is hereditary with respect to J-equi- valence. By theorem 14.1. this question finds an affirmative answer now. In fact, the answer is an easy consequence of some assertion (proposition 13.1.) concerning her- metic equilibrium distributions of age dependent GALTON WATSON processes (GW processes).

We say that an equilibrium distribution P is of standard type, if for each U E A the :mbedded y,,,-equilibrium P,,, is immigrative or hermetic. By proposition 14.4. an equi-

Math. Nachr. 160 (1993) 314

librium distribution P is automatically of standard type, if it is of first order. Example 14.3. shows, that this restrictive moment assumption cannot be removed.

Example 14.3. is based on the result of section 7. There we saw that subcritical age dependent GW processes sometimes admit a non-trivial hermetic equilibrium distribu- tion. This rather contraintuitive fact leads to the question, whether the “pathology” of section 7. can be carried so far that the stochastic branching dynamics of the age de- pendent GW process is a substochastic shift. This finds a negative answer by proposition 15.1. The proof was stimulated by the approach to corollary 3.9. in LIGGETT & PORT [4]. Moreover we use some notions from the monograph [2], chapter 4., and a general assertion (theorem 16.2.) which we prove in an appendix and which concerns the exist- ence of the characteristic J-invariant distribution Wp for an arbitrary equilibrium P in the case where the branching dynamics is a substochastic shift.

By theorem 15.2. the assertion of 15.1. is carried over to substochastic shifts on ar- bitrary discrete phase spaces A : Backward and forward transience of all individuals are shown to be equivalent in this case.

This paper is an immediate continuation of 161 in its intention and methods. Once again we emphasize that, as a rule, we make only tacitly use of the refined model carrying the information on genealogical relationships of individuals (cf. section 1 .).

13.

Let y be a fixed age dependent GW branching dynamics. Recall that in one time step, y decreases the remaining lifetime n of an n-individual by one (in case n > 1) and takes a 1-individual into a y,,,-distributed random population y on N. Let Q be a y-equi- librium distribution, and take Q as initial distribution of an age dependent GW process whose dynamics o is obtained by removing a certain non-trivial (i.e. being not a.s. empty) subpopulation r] from y. For short, we refer to a process with emigration.

If there is some steady stream of immigrants from infinity under Q and 7, i.e. if the y-equilibrium distribution Q has a non-trivial immigrative part Qimm in the sense of section 4., then our process with emigration will converge towards a non-trivial o-equi- librium: In fact, QU,”, is non-increasing (cf. the first step of the proof of proposition 13.1. below) and (cf. 4.3.) is bounded from below by Q,. If, however, there is no im- migration from infinity under Q and y, i.e. if (cf. section 4.) the y-equilibrium distribution Q is hermetic, then the process with emigration is doomed to extinction. This is the content of the following proposition.

Emigration from hermetic equilibria leads to extinction I

Proposition 13.1. Let y and u be two branching dynamics of age dependent GW pro- cesses, such that

“ , 1 ) 5 Y c l , ; ~ ( , ) * Y , l , .

QUm 18,.

Then for each hermetic y-equilibrium distribution Q we obtain

The proof of this proposition turns out to be lengthy due to its nontrivial core which concerns the case A,({ l}) = + 00. Before giving an exact proof, let us describe its ideas.

The assumptions on o and y can be described like that (cf. step 2.): The realizations

Matthes/Sigmund-SchuItze/Wakolbinger, Distribution 315

with respect to y(,) can be represented as x + q, where x - o(l) and q is not as . the zero measure 0.

For the first step of the proof we don’t even assume that Q is hermetic. As already indicated above, the process with emigration started off with Q “decreases” towards some o-equilibrium distribution L. Starting off the old dynamics y with the initial dis- tribution L yields a process which increases towards a y-equilibrium distribution V. Without any loss of generality we may assume that V = Q (step 1). This situation can be redescribed in terms of a random tree (i.e. a random population of paths) starting at time - a, part of which is white, describing the branching process with stationary distribution L, and part of which is black, namely the additional branches initiated by the offspring population q of white 1-individuals and developing further according to y. Thus, L and Q are recovered as the distributions of the white and the total population at time 0 (step 2). Denote by H the distribution of the black population above at time 0. Since black is constantly fed by white, the y-descendants of H die out in the long run (step 3). On the other hand, since any white branch has a positive probability of giving rise to a black descendance when hitting 1, and since such a hitting of site 1 has occurred not too far ago (thanks to hermeticity, which is needed here, and which L inherits from Q), one succeeds to show that H majorizes a certain non-trivial “part” L, of L, which is obtained by thinning the white “tribes”. Together with step 3, this forces L to be trivial (step 4).

Now we turn to an exact P r o o f . 1. We have

Q, 5 Q, = Q . So by 1.1. there is some L with

(*I Qu[nllL, Lu=L, L 5 Q . We get

L = L , 5 L y 5 Q y = Q ,

hence by 1.1. for some V

Ly(n]7 V , Vy=Vy V S Q *

Applying (*) to the y-equilibrium distribution V we get

L = L,[n] 5 V,,Lnls Q,[n11 L so that

Vdn] 1 L . Hence, intending to show L = 6,, without any loss of generality we may assume that Q = V.

2. According to our assumptions there exists a random ordered pair [ x , 171 of counting measures with

x - all],

Hence there is some k E IN with c:=iP.t&(q({k} > 0)> 0.

( x + vl ) - Y,1].

Math. Nachr. 160 (1993) 316

We introduce a refinement of the model by considering the marked phase space IN x {0, I } and a branching dynamics M acting as follows

B ( I ~ , ~ I ) : = balk- I , i ] for k > l , i = O , l ,

(x + v l ) 0 6, - %1,11) 3

x 0 6 o + v l 0 ~ 1 - % l . O l ) *

Moreover we define P := Q (U @ 6, E ( . )) .

Then we get for some W

In fact, let (ui) - Q. For all natural numbers n we may construct Pa,,, realization-wise by ‘colouring’ (i.e.%arking) wo. We colour all the y(, ,-clusters appearing in the evolution of (mi) according to the construction at the beginning of 2. Now we use this colour- ing-passing step by step from u-, to wo- to colour 0,. As n is increasing, we observe that the part of individuals in uo which is coloured with ‘colour’ 1 is growing mono- tonously, i.e. for n -+ co we have a realization-wise convergence. (We expect the colour 1 being dominant to comprehend the whole population, but this will be seen only later in the proof.) So there is such a probability distribution W with PdnI * W , and by theorem 4.7.2. in [5 ] it is possible to exchange weak convergence and clustering in our case. Hence we have the relation W, = W. ,,

Pa[,] =. w; w, = w.

3. Let (@i@6,+ !Pi@6,)- - W . As we saw in 2., this implies (Gi + Yi) - Q .

There holds for all natural numbers n -

Pa[n1(6(( ‘ 1 x IO) ) E ( ‘1) = Q,[n] . So @, -; L and

where the right hand side has to be formed according to the branching dynamics a (for which L is an equilibrium distribution). In view of the considerations at the end of l., any individual in some Yi has an ancestor in some !Pj, j < i.

(@i)-&,

We define

Then we obtain W(Y’, E ( . ))=:H .

(**I H y l n ] 1 6,. In fact, we may construct Hylnl as follows: We delete from Y o all those individuals which have no ancestor in (Qi) at times - n , . . . , - 1.

4. Until now we did not use that the y-equilibrium distribution is hermetic. Of course, this condition implies the hermeticity of the a-equilibrium distribution L, since L , 5 Q,. We intend to lead the assumption L =k 6, to a contradiction.

Let us denote by 9, the operator of independent thinning with survival probability c. Remember the definitions of the index k and of c at the beginning of 2. to realize the following relation

((9cL) ( (1 ,,..,n)@ E ( k] 5 _W(Yy, E( 1) = H for 12 2 k *

~~tthes/Sigmund-Schu~tze/Wakolbinger, Distribution 317

This means that we consider among the individuals in Yn those, which have the following property: The individual has an ancestor in @n, which, having reached the position one for the first time at some intermediate time n’, generated an individual at position k and of colour 1 at 11’ + 1, which is an ancestor of our given individual. The probability of any individual in @n to have this fate is c, supposed it has a remaining lifetime (po- sition) not greater than n.

If we replace y by a we get the inequality

( ( 9 c ~ ) ( , l , , . . . n ~ @ € ( . ) ) ) u l n - ~ ] ~ H for n 2 k . Theorem 2.11.6. in [2] tells us that we have the convergence

(J)cL)ulnl => Lc >

where L, is a a-equilibrium distribution. Now we use the hermeticity of L. I t implies that

((9cL) ((n + 1 ,...I @ E ( - k] 5 J)c(Tn -kL) for n 2 k . tends weakly towards 6, as n -, co, where T denotes the shift operator which was in- troduced in section 4. Thus we get

((19cL) ((1 ..... nl@ E ( . ))),[n - k] n- 3 cu L,<H.

I t is an easy consequence of the definition of L, that i t cannot be 6, unless L is the void population (which is exactly what we intend to derive). In fact, first we may assume without any loss of generality that c is the inverse m - ’ of some natural number m. Otherwise we could make c simply smaller. Then we choose rn colours and colour all the individuals of a realization @ of L independently and with equal probability for each colour. We apply now n times the clustering mechanism with branching law o to the coloured realization @, following the rule that all descendants of an individual get the same colour as this individual. So we get another coloured realization @‘, which is distributed according to L, if we neglect the colours, since L is a a-equilibrium dis- tribution. The totality of individuals in @’ with a given colour is obviously distributed according to (9,L)01nl. Since the number of colours is finite, it is impossible that all the subpopulations with a given colour die out, provided L =I 6,, showing that L, + 6,.

Now we have the desired contradiction which (remember (**)) is as follows: 6, + Lc = (Lc)oln] 5 HuInl 5 H + n ] 1 8,.

14. Equilibrium distributions of standard type

In this section we consider an arbitrary discrete model. For all equilibrium distributions P we define.

Aherm,P = { u E supp A,: PI,, is hermetic) First we give some augmentation of 9.1.:

Theorem 14.1. Let Pepinv, U E A and X:=A,, , . Assume thar A , ( X ) > 0. 7Ien P,,, is

- H (to each individual in @o there is some S E !I3 visited by its ancestors infinitely often) = 1 .

hermetic ifs X E a,,, and H := P I X I is ‘hermetic’ in the sense that

Math. Nachr. 160 (1993) 318

P r o of. 1. Assume PI,, is hermetic. By 9.1. we have X E '$I(,,). From (Qi) - it follows that a s . each individual in Qo has an X-ancestor. On the other hand it was shown at the end of the proof of 9.3., that a s . each X-individual in (Qi) has an a-ancestor. Then, of course, i t has even infinitely many a-ancestors. Hence H is 'hermetic'.

2. If PI,] is hermetic, then this is true for all P [ b ] , b ~ X \ { a ) . To show this we may assume without any loss of generality that P = PI,]. Assume that for some b ~ X \ { a ) the distribution PI, ] would not be hermetic, i.e. that the equilibrium distribution V : = P ( ( b ) ) is non-trivial. In view of P = P,,,,, we get V = V!,,,,, too, i.e. Vial is hermetic.

We use 13.1. and put y := K~,] and Q:= VIoI. We define o(l) in the following way: Instead of the random system of all direct a-descendants of an individual given at time zero, we consider just the subsystem of all those direct a-descendants, which have no b-an- cestor at positive times. By this set-up all assumptions of 13.1. are satisfied and we obtain

(*I Q , ~ n l l 60 *

If (Yi) - V , then by assumption with positive probability the subsystem x of those now b-individuals in !Po, which have no b-ancestors, is non-empty, i.e. we get

Obviously H is an @-equilibrium distribution for the clustering field a defined by the following modification of K:

H := _v(x E ( * )) * 6, .

@(,) := K(& (( . ) \PI) E ( c E A\W 9

a(,,) := 6, . Now o is nothing but the embedded branching dynamics at a with respect to a.

H . Then we have Let W denote the embedded a-equilibrium distribution at position a derived from

W 5 Q so that for all natural numbers n

6, * W = W,,[n15 Q,,[nl,

which is a contradiction to (*). So P [ b ] must be hermetic for all b in x\{a}. 3. Assume now that X is P-autonomous and that H:=P, , , is 'hermetic'. Let X E X and (Qi) - H . We denote by Qi,+ (resp. Q i , - ) the subpopulation of those

individuals in Qi, i E Z , which have infinitely many (resp. finitely many) x-ancestors. If we denote the distributions of (Qi, +)iez and (Qi , - ) iez by HI~,)l and H,,,,), respectively, then we realize (cf. the intuitive approach to 1.2. at the end of section 1.) that H,,,,,, H((,,)EP'"~. For two points x, Y E X we obtain by 2.

By assumption we know that H-a.e. each individual in (ai) has infinitely many C-

ancestors at a certain site C E X. Now from (**) we conclude that in that case it has almost surely infinitely many a-ancestors, too. Hence P,,,, which coincides with H,,], is hermetic.

By 14.1. Aherm,P consists of all J-equivalence classes of sites a ESUPPLI~ with hermetic P,,,. Obviously U ( p ) is closed under countable unions. Thus we get

(**I (Hl,X,l),,,,,) = 6" = (HI,,,l)(,X,) *

Proposition 14.2. AhermSP E (P E Pinv).

Matthes/Sigmund-SchuItze/Wakolbinger, Distribution 319

An equilibrium distribution P is said to be of standard t ype if for all a E supp A, the embedded yI,l-equilibrium is either hermetic or immigrative, i.e. if P,,, is immigrative for all sites c outside of Ahcrm,p. This property is not shared by all equilibrium dis- tribu tions.

Example 14.3. Let be y the branching dynamics introduced in section 7. describing a subcritical age dependent GW-process and let Q be a non-trivial hermetic y-equili- brium distribution. We denote by V the immigrative y-equilibrium distribution with

V , = d,, where p:= 1 a,, k > O

which exists due to 6.1. Let A : = Z + ,

K ( , ~ : = Y ( ~ j E N ,

K(o) :=ado+dl *

We may interpret Q as a K-equilibrium distribution with the property A Q ( { O } ) = 0. If now ( x i ) - V and ifwe denote by W the distribution of the subsystem of those individuals in xo whici are not “fresh” from the source in the sense that they have at least one l-ancestor, then we obviously have

4+6do* w , since the (0)-emigrants now substitute the distribution Vm. Then by ddo * W we find another K-equilibrium distribution. Putting

PI= Q *(ado * W ) ,

Ahcrm,p = {0} ; P I , ] is not immigratiue . we get a K-equilibrium distribution with the properties

Note that P is ‘hermetic’ in the sense that P-a.e. each individual in Q0 has infinitely many (0, I}-ancestors. g

The equilibrium distribution P introduced in example 14.3. is not of first order. This is not by chance, since by 8.3. we have

Proposition 14.4. Any equilibrium distribution P of first order is of standard type, and

A P ( A b C , , . , A C ) = 0,

C : = { U E A : 1 J ~ ” l ( a , { u } ) = +a}. where

n S O

15. The subcritical counterpart to proposition 5.3.

Just as in section 13. we will use the notions and notations of section 2. To begin with, we recall that by 6.2. for subcritical y each y-equilibrium distribution

of first order is immigrative, whereas the example of section 7. shows that this is no longer true if we admit arbitrary y-equilibrium distributions. Therefore it is not obvious

Math. Nachr. 160 (1993) 3 20

at all that any equilibrium distribution is immigrative in case of y being a proper substochastic shift. The proof of this assertion is more difficult than for its critical counterpart 5.3.

Proposition 15.1.1s the branching dynamics of an age dependent GW process y j i ~ l f 1 ~

Y ( I ) ( X ( N 5 1) = 1 ; Y(I)(X = 0) > 0 , then each y-equilibrium distribution is immigrative.

Proof . 1. We have to show that any hermetic y-equilibrium distribution Q is trivial. By 16.2. there exists the characteristic J-invariant distribution

“Lh=: V . Now in view of 4.7.4. in [23 the triviality of Q is equivalent to the assertion that the

2. For abbreviation we set .I,-invariant distribution V coincides with 6,.

qk:=Y(l)(X({k}) ’ O) ( k E N ) . Obviously the statement of our theorem is valid in the case that all q k are zero. So we may assume from now that

k > O

Let us denote by d the smallest natural number such that

If d is greater than one, then by putting

Q,:=Q(( 2 @(Is + k d } ) Q ~ ( . ) ) k 2 I

we get for s = 0,1,2, . . . , d - 1 hermetic y’-equilibrium distributions, where

Y;1):= c q k d ’ 6 6 k . k > O

Since Q is trivial iff all of the Qo, . . . , Qd- are so, we may assume from now that d = 1. 3. In the appendix we define for an arbitrary branching dynamics K the notions of

a h--process and of the set ,?Ir. Let us make use of these notions in the following. We say that a y-process H is hermetic, if a s . each individual in ( x k ) - H has a 1-

ancestor. If we assume additionally that H is of first order, i.e. that

(%I:= (4f(XpE(.))) E ,;;Ir 3

then we get

(*I v k ( { m ) ) = c V k - s ( { l ) ) q s + n i - l ( k ~ i Z ; m E N ) . s > o

We introduce in the appendix the kernel E(.,.) and the distribution _V, which are

If we take into account that together with Q the y-process K((@J,( .)) is hermetic, defined in terms of the branching dynamics and the given equilibrium distribution.

too, - Q-almost surely, then we realize that

- V((v,) belongs to ,;:Ir andfulfils (*)) = 1 .

~~~thes/Sigrnund-S~hultze/Wakolbinger, Distribution 32 1

4. Jf a family ( v k ) in N;;“ satisfies the condition (*), then we get for u k : = v k ( { l } ) , k E Z , fie convolution equation (**I u k = uk-jqj ( k E Z ) .

BY theorem 3 in [I] all non-negative solutions (uk) of (**) are given by uk = s k . c, where j > 0

2 0 is arbitrary and s > 0 fulfils 1 S - k q k = 1 .

k > O

Since 0 < 1 q k < 1, we obtain 0 < s < 1, and hence uk tends to 0 for k + + co. Hence by 3. k > o

- V ( v k ( { l } ) T ‘W O) = .

In view of the shift invariance of _V this implies - V(v , ( { l}) = 0 for all i EZ) = 1 .

Again we consider an arbitrary discrete model. If the stochastic branching dynamics K is a substochastic shift we may apply 15.1.

to get the following assertion: For an arbitrary equilibrium distribution P the property

(i.e. the ‘immigrativity’ of P in the sense of theorem 8.1) is equivalent to the property

In other words

Applying 3. once again, we get _V(v, = o for all ~ E Z ) = 1, i.e. V = 6,.

- P(al1 individuals in Qo are backward-transient) = 1 ,

- P(al1 individuals in Qo are forward-transient) = 1 .

Theorem 15.2. I f K is a substochastic shiji, then for any P in Pinv the following assertions are equivalent a) - P-almost surely for each X E B all individuals in have only finitely many X -

ancestors, b) P-almost surely for each X E 23 all individuals in Qo have only finitely many X-de-

scendants. P r o o f . 1. By assumption K describes a substochastic shift according to the sub-

stochastic transition matrix

( P a . d : = ( J ( a , {b)))a.bEA . So all

15.1. this implies a).

O L E A , are substochastic shifts. 2. Let us assume that P fulfils b). Then all the

3. By 8.1. b) is a consequence of a).

C I E A , are subcritical. By 8.1. and

16. Appendix: An existence theorem for the characteristic J-invariant dis- tribution

Just as in section 1. we will consider now the basic model in full generality, i.e. without the discreteness assumption.

21 Math Nachr.. Bd. 160

Math. Nachr. 160 (1993) 322

Let, for all n EZ, Fn denote the a-algebra on M" which is generated by the mappings ( @ i ) H @ j ( X ) ( j s n ; X E % ) .

Let further F;_m:= r'l .Fn.

neZ

A distribution H on .F:=ID1@'" is said to be leji tail trivial if it takes only values 0, 1 on 9-=.

A distribution H defined on F is said to be a K-process if it is a MARKOV process (@i)isz of counting measures with the transition kernel @I-+ K(@) given by the branching dynamics. Hence a K-process H is shift invariant iff

H(@,E(.))=:PEP~"' and H = p . To each ic-process H we may assign the 'refined' KO-process H" in a canonical way. Especially we have

(PJ" = p" ( P E Piny. We do not intend to demonstrate this here in detail.

A K-process H is said to be o j first order if v ~ : = A ~ ( @ ~ ~ ( . ) ) E N ( iEZ) .

( v ~ ) ~ ~ ~ E N ; " " : = { ( ~ ~ ) E N ~ : y j + , = y , * ~ for a l l j ~ Z } . In this case H is said to be a K-process with ihkensity sequence (vi) . Then we obviously have

Let us denote by % the a-field of subsets of the set N of all locally finite measures (on 9I) which is generated by the mappings v H j d v , where j is bounded, continuous and with bounded support. We denote by => the weak convergence within the set V of all probability distributions on %. A distribution VEV is called J-invariant, if I/-almost surely we have v * J E N and

v(v*JE(.))= v is valid.

family ( v ~ ) ~ ~ ~ of locally finite measures such that

The distribution _V is uniquely determined by these two demands. Obviously _V is shift invariant.

By 4.1.1. in [2] there exists a kernel 5 from [M', 9- ,] to [M', 91 with the following properties (i) For all (@JE M", Z((Qi), ( .)) is a left tail trivial rc-process,

(ii) For all K-processes H, (@JHK((@J, (.)) is a version of g(( .)l.F-m).

To each J-invariant VEV we may assign (cf. [3]) the distribution _V of a random

- V(N?")= 1 ; y(voE('))= v.

Thus for all P E Pin' we get the integral representation

P = jK((@iL ( . )) E(d(@i) ) . Under the assumption, that - P-almost all K-processes K((Oi), ( .)) are of first order, we denote the distribution of ,4B((Qi),xoE(.)) by "lrp and call i t the characteristic J-invariant distribution of P. By 4.7.4. of [2] P I + "w, is one-to-one into the set of all J-invariant

~atthes/Sigmund-S~huItze/Wakolbinger, Distribution 323

distributions in V. I t Is easy to see that in case of the existence of Wp the distribution qp is the probability law of the family (nli((9,),xk.(.)))ksz.

The existence of Wp is obvious for equilibrium distributions P of first order. In general we have

Proposition 16.1. For an equilibrium distribution P the distribution Wp exists ifs

-

P ( @ * P I E N ) = 1 ( n = 1,2, ...) and, for all X E 8, { P ( @ * JLn1(X) E ( . ))},, is relatively compact with respect to the weak topology in the set of all probability distributions of random non-negative real numbers.

Proof . 1. Consider any left tail trivial distribution H of a tc-process. If (x,) - H and X E 8, then almost surely we get

(x - k * J [ k l ) (X) - x O ( x ) (d(xn)). (*I k + ai

In fact, let us first consider the special case where the right hand side of (*) is finite. Then

That means that the left hand side of (*) constitutes a reverse martingale and con- sequently it is as . convergent towards some 9- ,-measurable random number ( with expectation J x o ( X ) H(d(xn)) . By assumption C almost surely coincides with its ex- pectation.

(x - k * J‘kl) = E H ( X O ( X ) I g- k ) ( k E Z + 1.

If the right hand side of (*) is infinite, we define for k , m E Z + t k . m := (m, x O ( X ) ) I @- k ) 9

and as above we almost surely get for each m E Z+

Since almost surely

we have almost surely

r k , m J (m, x O ( X ) ) H ( d (xn)) .

(1- k * J’A1l) 2 t k , m >

lim (x - k * Jikl) (X) 2 sup J min (m, xo(X) ) H(d (x,)) = + 03 . k q m = 1.2.. .

2. Now let P be any equilibrium distribution, X E 8 and ( X k ) - P. As we have seen above, P has a representation as a mixture of the - K ( ( @ J , ( . ) ) , (@& M”. So by 1. we almost surely get for some r x

and there holds (x- k * J[k l ) (X) < x >

- P((x- , * J r k l ) ( X ) < + cc for all k E Z , and t x < + co) = p(J v ’ O ( W & ( ( x i ) , ’ ( V ’ n ) ) < + m).

If we take into account the shift invariance of P, by 4.1.4. in [2] we may continue (for any m E Z + )

-

= - P(J V I ~ ( X ) K((xi), d(vjn)) < + a). 3. In view of 2. for any equilibrium distribution P the assertion

“for all X E !.I3 the random numbers q k : = ( x - k * J lk l ) (X) are a s . finite for all k E Z +, and the sequence of probability distributions of these random numbers is relatively compact”

Math. Nachr. 160 (1993) 324

is equivalent to the condition

- P( 1 vi,,,(X) g((xi), d(y, ,)) + co for all X E 23 and all rn E Z,) = 1 . That is exactly what we had to show.

Based on 16.1. we obtain the existence assertion

Theorem 16.2. I f the branching dynamics K constitutes a substochastic shijt, then for

Proof . Consider some natural number n, some L > 8 and some 8 E M such that all P€Pinv there exists the distribution W p .

@ * J["](X) > L . Represent @ as 1 6, and write p j for JLnl(xj, X). So jd

r:= 1 p j > L . j d

Now observe that pi is the probability for an individual at position x j to have an X - descendant n times later, since K is assumed t o be a substochastic shift. Then by CEBYSHEV'S inequality we get the estimate

K[$(x(x) > ~ 1 2 ) 2 K{$)( (x(x) - r)' < r2 /4 ) 2 1 - 4r -* p j ( l - pj) j d

2 1 - 4 r - ' 2 I - 4 ~ - ' 2 1 / 2 .

P(@ * J'"](X) > L) 5 2 J K!$)(x(X)> L/2) P ( d 8 ) = 2P(@(X) > L/2) . This last expression does not dpend on n and it tends to zero as L tends to infinity. Hence the family {P(@ *J"$Y)E( .))},,,, is relatively compact as a family of distribu- tions of positive random numbers. In view of 16.1. we are through.

Therefore we obtain, taking into account that P E Pin",

References

[ I ] G. CHOQUET, J . DENY, Sur I'equation de convolution p = p * u, C.R. Acad. Sci., Paris 250 (1960)

[2] A. LIEMANT, K. MATTHES, A. WAKOLBINGER, Equilibrium Distributions of Branching Processes,

[3] T. M. LIGGETT, Random Invariant Measures for MARKOV Chains, and Independent Particle Systems,

(41 T. M. LIGGETT, S. C. PORT, Systems of lndependent MARKOV Chains, Stoch. Proc. Appl., 28

799-801

Akademie-Verlag, Berlin, and Kluwer Academic Publishers, Dordrecht (1988)

Z. Wahrsch. verw. Geb., 45 (1978) 297-313

(1988) 1-22 IS] K . MATTHES, J . KERSTAN, J . MECKE, Infinitely Divisible Point Processes, Wiley, Chichester (1978) [6] K. MAITHES, RA. SIEGMUND-SCHULTZE, A. WAKOLBINGER, Equilibrium Distributions of Age De-

pendent GALTON WATSON Processes 1, Math. Nachr., 156 (1992) 233-267

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