Some results for non-supercritical Galton-Watson processes with immigration

Some Results for Non-Supercritical Gabon-Watson Processes with Immigration A. G. PAKES Mathematics Department, Monash University, Clayton, Victoria; 3168, Australia

Communicated

by Richard Bellman

ABSTRACT This paper considers a population of objects reproducing according to a non; supercritical Galton-Watson process and which is augmented by an immigration component. Some results on the classification of the states, convergence rates for the transition probabilities and limit theorems are obtained when certain moments are infinite. Some of. the limit theorems are obtained for a time varying and a state dependent immigration component.

1.

INTRODUCTION

We consider the well known Galton-Watson process with immigration (G.W.I. process) [ 1, p. 2621. This is a Markov chain (X,,; n = 0,1, . . . } whose state space is the non-negative integers and whose transition probabilities are given by pii = coefficient

of sj in h (s)(f(s))i

where h(s) = ~,~_,,hjsj and f(s) = Z,?_Opjsj are probability generating functions @.g.f.‘s). We interpret X,, as the size of a population at time n in which an individual in the population at time n producesj progeny with probability pj immediately before its death at time n + 1, and in which j immigrants enter at n + 1 with probability 5. Individuals reproduce independently of each other and of the immigration component. Assuming, as we shall in what follows, that 0 < h,,, p. < 1, it is known [ 181 that the state space can be written as S u E , S n & =$I where S is infinite, irreducible, and aperiodic, and contains (0); and that the n-step transition probabilities { pijn)} satisfy p0(“)=O (Jo&, all i, n>O). Ifp,>O, then & is empty. Thus we can, and shall, take S as the state space of {X,}. MATHEMATICAL

BIOSCIENCES

24, 71-92 (1975)

6 American Elsevier Publishing Company, Inc., 1975

71

72

A. G. PAKES

It is of some interest to classify S , a task not yet completed in full generality. Let o=f'(l -). When (Y> 1, S is transient [7, lo]. When (Y= 1, /3 = h’(1 -) < co, it is known that S may be transient, null or ergodic. More specifically, if y =f”(l-)/2 is infinite, S can be ergodic [ 171, and if y < co, (I = p/y, then S is transient (null) if u > 1 (< 1); and if u = 1 and zhj jlog+j < co, Zpjjzlog+j < cc, then S is null [13]. More generally, Zubkov [23] has given a necessary and sufficient condition for n 11-recurrence when u = 1. When (Y< 1 an early and important result [7] in the field was that S is ergodic iff Z hj log +j < co. One aim of this note is to obtain some information for cases not covered above. We always assume (Y< 1. In particular we shall state a sufficient condition for transience or null-recurrence and show that when (Y< 1, Z hj lOg+j = co, S may be null or transient. When (Y= 1, p = cc, we shall see that S may exhibit any of the three modes of behavior. We obtain some asymptotic expressions for pp) and mention some consequences. In particular we obtain (a simplification of) Zubkov’s criterion. A number of limit theorems are known for {X,} and { Y,}, where Y,=X,+**. +X,, when suitable moment conditions hold. Heathcote’s theorem [7] cited above is regarded as such a one. In Sets. 3 and 4 we obtain some analogues for these results under different assumptions. In particular we obtain some limit theorems for {X,} when a < 1 and S is not ergodic. Pinsky [ 161 has stated, without proof, analogues of these results for continuous state branching processes with immigration. These results are examples of non-linear limit theorems. Such a theorem also occurs in the explosive (a = co) Galton-Watson process without immigration [20]. Recently such a result has been obtained by Foster [5] for a critical GalfonWatson process with an immigration component modified so as to act at time n + 1 only if the population size at n is zero and under “regular” moment assumptions. Foster states a linear limit theorem for a special form of immigration distribution which ensures that its mean is infinite. In Sec. 5 we shall prove an extension of this result and exhibit the density of the limit law as a mixture of a certain two parameter density. Finally, in Sets. 3 and 4 we give some limit theorems in which we drop the assumption that the number of immigrants entering at time n, I,,, has a distribution independent of n. The methods follow those in [6].

2.

ON THE CLASSIFICATION

OF S

When (Y< 1 it is known [18] that {X,} has an invariant measure { b; j E S } which is unique up to multiplicative constants. We take p0 = 1. The working in [ 181 may easily be extended to show that

.

GALTON-WATSON

so that the asymptotic fn+I(s)=f(fn(s))

73

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form of p,lnj follows from that of p&j). Let f&s) = s and

(n=O,L...)

andf,=_tXO).

l-hen [71

n-1 fl h(f,).

P$‘=

(2)

m-0

Since f”-+l, p6;;+‘)/p6;;)+1 ( n+m), and an appeal to Raabe’s theorem the convergence of positive term series yields THEOREM

1.

Suppose

lim,,,n(l

on

S

is

- h(f,))

not

ergodic.

Then

S

is

transient

(null)

if

> 1 (< 1).

Remark. The principle result of [24], shows that the existence of the limit, 0 < p < cc, in Theorem 1 is necessary and sufficient for the representation p&j= nvPL(n),

where L(a) is slowly varying (S.V.) at infinity. The next theorem represents p# THEOREM

in terms of a S.V. function.

2.

For each 5 > 1 there is a non-increasing function such that p$‘=

Furthermore Remark.

S.V. at injniq,

(n=O,l,...).

qm

cl (X)N e, (x) (x-+00;

In the subcritical

C,(e),

5, v > 1).

case it is most natural

to take { = CY-‘.

Pro05 On setting

f!,(x) = pk("'l, we have the representation and observing that

log+ x 4(x) = logs

(x > O),

above. For any h > 0 take x > 1, so that Ax > 1,

sup 14(b)-

x>l

q(x)1 < 00,

it follows that tZ, (AX)/ E, (x)+1, hence the S.V. property. 1 > v > 1, monotonicity and the S.V. property show that qs+

G(vY)

(.Woo),

so

W)+(x’+K),

Supposing

that

74

A. G. PAKES

where K=(log{)/(logv)1 >O. Thus the second part of the theorem will follow once we show that C, (x’+~)C, (x). This follows from Theorem 2 of [3] on observing that for A,= 5 and T(x)= xK,

(gg - l)logT(x)~o is equivalent

to

+&

(“-‘)l/pg”)l)+o

and the latter statement is a consequence and the principle result of (241.

(n-+co), of the first part of the theorem

Let {Z,} denote the simple Galton-Watson process with offspring p.g.f. f(s) and Z,,= 1. When a< I it is well known [18] that E(sq]Z,>O)+Q(s) (n-+oc), a p.g.f. of a random variable Z, and that Q(e) is the unique p.g.f. solution of

Q(.f<~>>=~Q(s)+l-~2

Q(O)=O.

Furthermore c- ’ = E(Z) is finite iff Zpjjlog+j < co. Let +(.) be the inverse function of 1 - Q(1 -s). Recently Seneta [25] has provided the following answer to a long open problem [l, p. 631. THEOREM

S

I-f,=cw~(a”) where $(s)=(p(s)/s

is non-decreasing,

S.V. at the origin and &c (sJ0).

It is not without interest to observe that Theorem S can be eventually deduced from Theorem 2. For the case p, > 0 the essential step is to show that 1 -f,-Q’(0)cPcn where c, = P(XL =0(X6=0) and {Xi} is a G.W.I. process with offspring p.g.f. f(s) and immigration p.g.f. $(s)=f’(s)/cr. The proof, suggested by [ 1, pp. 38401, proceeds by defining k,,(s)=(f,(s)l)/cr” and observing that k~(s)=E(sX~]X~=O). It is shown in [12] that Z~,,[l;(f,(s))-_(f,)]< cc, so that with c, = k;(O), K(r)/c,?

Us)

where U( .) is a generating function is the unique such solution of f’(s)xHr))

< 00

(O
(of the invariant

= au(s),

measure of {Xd}) and

U(0) = 1.

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Clearly Q’(s) satisfies this equation, Monotone convergence now yields

1-f,(s)

-5

~%I

so U(s)= Q’(s)/Q’(O)

l-Q@>

Q,(o) J~~drtjlU(s)ds= s

C”

,

s

and the assertion follows. A suitable A version of Theorem 2 for the akin to the spirit of [25] as follows. the invariant measure of {X,}. This

reformulation will cover the casep, = 0. case cr < 1 can be obtained in a manner Let U(s) be the generating function of is the unique solution of U(0) = 1

U(s) = h(s) VU(S))> and 9 (s) = U( l-s)

and Q’(0) >O.

is S.V. at s = 0; see [26]. Iteration p&j’= l/U(h)=

and Theorem

S yield

l/~(cu~(cw”)).

It is easily seen that 9 (@(s)) is S.V. at s = 0. A similar approach may be taken for the case (Y= 1. This requires the following lemma, which complements Theorem 2 of [26] and whose proof is suggested by its proof. LEMMA When

1. a-

1,

where +( .) is decreasing and S.V. ProojY The equation for the stationary measure of the Watson process can be put into the form e9 (s)= ‘?P(f(s)), 9 (a) is strictly increasing on [0,1) and has a power series non-negative coefficients, and ‘3’(s)tco (sT1) [26]. Writing and F(s) = 1-f( 1 - s), this equation becomes

critical Galtonwhere 9 (0) = 1, expansion with P(s)= 9 (1 -s)

eP(s) = P(F(s)) and iteration yields the asserted representation +( 1) = 1, $( t)$O (I-+ co), and +( .) satisfies

with cp(t) = P -l(t).

4eY) = F(+(Y)). Thusif

l
@(es)_ F(+(y)) ,*,

44s) whence the S.V. property.

NY)

Clearly

76

A. G. PAKES

Returning

to the critical G.W.I. process, it is clear that p&‘= l/A(e”),

where h(t)= U(1 -+(t)). the functional equation

Clearly A(l)=

1, A(t)Tcc (ttco)

and A(*) satisfies

where H(s)= h(1 -s). It may now be deduced as above that A( .) is S.V. at infinity. We now prove an asymptotic result for the case (Y< 1 which can be used to provide information on the rate of decrease of { ~6;;)) when p = co. The following condition will be used: A. CpJlog+j<

co.

We recall that Condition A is equivalent to the requirement 1 --f,-COI’ E(Z,(Z, > I), we have c < 1. We let (O
3.

Let a< 1, zjhjlog+j=

cc and {=(logcy-l)-‘.

Then as n+co,

‘--a” logh( Q -l(s)) ds I-s If condition A holds, (4)

Proof. We first prove (4). The condition is equivalent to the requirement that

s

l l-h(s)

-T-sds=

on the immigration

distribution

co.

0

Denoting

the coefficient

of K, by e,, we have n-1

en

(n m=O

h(l-cam)

1-‘=expS,,

77

GALTON-WATSON PROCESSES

where

w n-l

S,=

‘-cam+’log/r(s) ~ds-log

s

=WJ n-l

m+l

m-0

Now -log/t(s) satisfies

(h(l-cam)) 1

1-m”

m-0

logh(l-

ccw*)df-logh(l

-Cam)

m

is positive and strictly decreasing,

I

.

so the mth summand,

o,,,,

“+I)-logh(l-UP).

O
Thus {S,) is non-decreasing and uniformly bounded, so exp S,, converges to a positive finite limit. To show that p# (~,-,‘,h (l- CO!‘“))-’ converges it suffices to consider the series

s=

5

[h(l-cq-h(f,)]<~.

m=O

Since (1 -f,)(w -“&, the summands are positive. In the other direction, have (1 -f,+ ,)CX-m-+~~ < c, since c > 0. Thus there exists M such that

we

(m>W,

1-ca!“
m=M+I

m=O

and the telescoping series sums to 1- h(f,+ ,). The proof of (3) is very similar to the first half of that of (4). Denoting the coefficient of K by d,,, we have d,,/p@=expD,, where logh(Q -‘W) 1-S

ds_logh(f

)

m

Setting s = 1 - cr’, it is easily seen that the integral becomes I;+’ logh( 1 -+(cu’))dt, ,where +(*) is as in Theorem S. The proof is now easily completed. It is clear that Theorem 3 holds when ~jh,log+j< cc, but it is then no longer interesting, being subsumed by Heathcote’s theorem [7]. Assume that A holds, and consider the immigration distribution I$= cj-‘(logi)-”

(j=

1,2,...,

1
78

A. G. PAKES

with C chosen so small that h,,>O. Zjh, < cc and Zh,log+j= co. Clearly

$ h,-C(Sso an Abelian

The constraint

I)-‘(logj)‘-g

on S ensures

that

(j+,),

theorem yields l-h(s)-C(i3-l)-‘(-log(l-s))‘-6

(STf).

We thus obtain Gl=-”

n(* -hull)-

(6-

l)(loga-‘)

6-l

so that if 6 < 2, Theorem 1 implies the transience of 5 , while if 6 = 2 we can choose C so small that C/(6 - I)(logcu 1)6-’ < 1 and S will be null. If 6 < 2 we have I,=

s

‘-ca” 1 -h(s) ____ 1-s

I-c

2-6

&-

(2_&_

*> (nloga-1)

and writing (4) yields

for n > N(E) and E arbitrary but fixed. Thus when the limit p in Theorem 1 is infinite, we can obtain a situation where geometric ergodicity fails to hold but pS;;) decreases faster than any regularly varying sequence. Turning now the critical (a = 1) process with immigration we prove LEMMA

2.

If P= CO,y < CO,S Proof:

is transient,

and if /3< o3, y= CO, s is recurrent.

In the first case we have n(l-

and Theorem

h(f,))

> n(l -f,V’(f,>-+~

1 yields the result. To prove the second

part it suffices to

GALTON-WATSON

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show that n(1 -f,)+O.

To see this let p(s)=f”(s)/2

and observe that

(l-~+,)-‘~(l-~-(l-f,)2P(JJ-’ P(f,)

=(1-f,)_‘+ 1 -Cl

>(l

-f,Mf,)

-JJ’+df,>.

Thus [n(l -f,)]-‘> n-‘[1 +Z”,=,,p(f,)]+o. It was shown in [6] that S is ergodic iff Z=

s

l ____ l-h(s)

o f(s)-s

ds
If @ E >O (x >O), then I< co. The following example shows that 5 may be null when y = 00. Let g(x)=(x(log~)~)-6 (x > 2, 0< 6 < 1). It is easily seen that g”(x) > 0, so we obtain an offspring distribution by defining Pj=c[g(j)-2g(j+l)+g(j+2)1

andp,=p,

(j=2,3,...)

and 2p,+C(g(2)-g(3))=1.

Let

so that %=1-P,,

(j=

m,=C[g(j+l)-g(j+2)]

1,2,...).

Thus m(s) is a p.g.f. (that is, CY = 1) if C=( g(2) + g(3))-‘. y(.r)=(l--(s))/(l-s) so that f(s)-s=(1-s)2y(S) and y,= Cg(j+2)

(j= 1,2,...).

Thus

y(s)--K(log(l

-S)-‘)1-s

(O
and we obtain

s cc

z>PK(l-E)

K

dx

x(logx)‘-6

=cO

(O
1)

Finally

let

A. G. PAKES

80 if K is large enough. Hence The following THEOREM 1. a =

S is null.

result is the critical analogue

of Theorem

3.

4.

1, then with the notation of Lemma p$)-

‘” logh(l-+(Y))

Kexp s

dv

(&Co).

Y

1

Zf also x=ZZ&~{(~ - h(f,))(l -f’(f,J>} p@-

1,

< ’

K, exp

00,

then

logh (s) ds

s ef0-s. Proof: The proof of the first part is almost the same as that of (3); just let y = e’ and use Lemma 1. The proof of the second part is a simple refinement of the argument used to establish the Foster-Williamson [6] criterion. We only need to show that {S,,}, where

converges. Denoting the integral by Z,,, and observing 1/(f(s) - s) are increasing, we have

Thus S, + , >S,,>O,

and

S,
that -logh(s)

n-1 (logh(f,+,))(f,+,-2f,+,+f,) c f In+2 -fm+, m=O

equals

- (logh(fm))(l -f’(L)) f’(L) =o{[l-h(f,)l[l-f’(f,)l}, and the theorem follows.

(fm< L
and

GALTON-WATSON When

Y
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PROCESSES

1-f,-(yn)-’

[l, p. 191 so x
if p
Furthermore

-logh(s)=l-h(s)+0(1-h(s))2, and Sh(l -h(s))2/(f(s)-s)2ds<

co. With a=P/y,

s

we have

_ %l-+_(l--hwds

’

f(s)-s

0

and it is not difficult to see that the second and third integrals on the right converge (n+co) if Cpj210g+j and ZQlog+j< CO,respectively. Hence p$-K,n-” a

result obtained If

(n-co,

by somewhat

different

0-c K, < co),

means in [ 141.

l-@)=(I-s)?

(O
where L(e) is slowly varying at infinity, f < 6 < 1 and ,!,(a) is sufficiently “regular”, p$)-Kexp(

Zubkov

l),

then it can be shown

that if

- ~~‘(1 -S)-ln’-GL(n)).

[23] has shown that when (I = 1, S is null iff n-l

00

c

Y

--(I-h(f,))

n-‘exp c[ m=O

fl==l

‘+Ym

1

=co.

(This is assuming that the a(.) which occurs in the statement of his result is the same as that implicitly defined in the proof of his Lemma 4 and not as explicitly defined in the proof Lemma 3.). Now n-l

c

-

Y

m=O I+Ym

=logn+O(l),

82 so

A. G. PAKES

his criterion

simplifies to co

S is null

iff

(

= fl=l

This result is a simple consequence the integral therein equals

-

s

THEOREM

Let M(l/(l

= 00. 1 5, since

n-1

x

(l-h(f,))+O(l).

m=O

f(s)-s

The proof of the equality bounding technique used The following result Williamson criterion and

h(f,))

of the second part of Theorem

fnp++(1)=l-h(s)

o

n-1

exp - *s,(l-

is similar to that of Theorem 5 but employing the in [6]. follows directly from Theorem 1, the FosterSlack’s Theorem [21].

6.

(Y=l,

l-h(s)=(l-s)*

L(l/(l-s))

and

wfrere O< 6, v < 1 and L(e) and M(.)

-s)),

f(s)-s=(l-s)‘C’

are slowly varying. Then

(i) if 8 > v, S is ergodic; (ii) if S < v and S(1 + v) > Y, S is null; and (iii) if 6( 1 + v) < v, S is transient. 3.

LIMIT

THEOREMS

FOR THE POPULATION

SIZE

Recently Pinsky [16] has stated, but not proved, a limit theorem for a subcritical continuous time and state branching process with immigration where no limiting distribution exists. We now prove the analogue of this result for the present situation. THEOREM

7.

Let ‘-e-X logh(Q H(x)=

s0

-l(s))

~ (O
l-s

and m(x) = eWW)

(l
and define m(e) arbitrarily on [0, 1) but so that it is continuous and strictly increasing everywhere. Let 4(x)= l/+(x-‘) (x > l), and define q(e) on [0, 1) so that it is increasing and continuous. Assuming that

ff(x)-+m,

xH’( x)+0

(x-+co),

then

m(F’(Xn)> m(a_“)

Lx

GALTON-WATSON

83

PROCESSES

where P(X< and { = 1/logo -

u)=ul

(OG u< 1)

‘. If in addition Condition A holds, m(xn> m(a-n)

d

+X.

Proof. Since ‘k,(s)=E(s”)=P,-,b h(f,(s)), it follows as in the proof of where {s,,} is a sequence of Theorem 3 that \k,,(.r,)/~,(s,J+l ( n+cc), positive constants strictly increasing to unity and @,,(S)=exp

/

e’[logh(l

Q(~))))]dt.

-+(a’(l-

(5)

We have used the fact thatf,(s)= 1 -$(a”(1 - Q(s))); see [25]. Substituting r = 1 - a’(1 - Q(X)) into the integral at (5) and writing M(x) = exp H(x), we now see that

*k,(d-

M( -log(l-

’

Q(G)>)

I

M(-nloga-log(l-Q(,)))

(6)

where s,, = Q -‘[exp(- e/K(n))] and K(n)= m-‘(vm(a-“)) (0< Y < 1). Note that m(x) is strictly increasing and m(x)+oo (x+00). Thus the inverse m-‘(x) possesses the same properties. Clearly M(x)Tcc, and since xM’(x)/ M(x)= xH’(x), it follows that M(s) is S.V. The uniform convergence theorem for S.V. functions [2] thus shows that the numerator in (6) is --M(logK(n)) Similarly

the denominator

= m(K(n))

is --m(cy-“K(n)).

= vm(a-“) We shall show below that

m(a-“K(n))--m(a_“). Assuming

(7)

this to be true, we have shown that

E[(l-+(l-exp(-B/K(n))))Xn]+vi

(O
1).

The right hand side is the Laplace-Stieltjes transform (L.S.T.) of a distribution having an atom vT at the origin and assigning zero probability to all intervals bounded away from the origin. Observe also that +( I- exp( - ,9/ K(n)))-((B/K(n))L(I /K(n)), where L(s)=+(s)/s is S.V. at the origin. Un-

84

A. G. PAKES

iform convergence

of, and the continuity

theorem for, L.S.T.‘s yields

Putting x = 1, the left hand side becomes P { m($(XJ) d vm(a-“)} and the theorem follows. When Condition A holds, L(s)& > 0, and setting x = c yields the second part of the theorem. To prove (7) first note that "WQ-'(WO,, t is S.V. Writing T(x)= m-‘(vm(x)), any AE(O, 1) and

we note that xAT(x) is increasing

for

o< T(x) 1, a mean value theorem for integrals yields O<

1og7-(x)=0[(1ogx)(-1ogh(Q-‘(1-1/x)))]

However, H’(x)= -logh(Q-‘(l-e-*)), (logx)( - logh( Q -‘( 1 - l/x)))+O. 2 of [2]. Remarks. Alternative For example, if Condition

and so xH’(x)+O is equivalent to Equation (7) now follows from Theorem

hypothesis on H(a) yield different limit results. A is fulfilled and H(x)=logx, then

P(~
(O
and if H(x) = xAr(x) where r( .) is slowly varying at infinity, P

1%X” n’/“-A+,(n)

(O
l),

then

+exp( - Ax-c’-A))

(0 < x < co),

where p( .) is slowly varying and satisfies pA-‘(x)r(x

The proofs remarks.

‘/(I-A)p(x))+l.

of these results follow directly

from (6); see [16] for similar

GALTON-WATSON

PROCESSES

85

We now turn to the critical case for which a number of limit theorems are known. For example if p, y < cc, Foster, Seneta [ 1, p. 2651 and Pakes [ 131 have independently shown that {X,/n} converges in law to a gamma distribution. This result has been extended by Foster and Williamson [6] to the situation where the immigration distribution depends on time. More precisely, let Z,, (n= 1,2,. ,.) be the number of immigrants entering the population at time n, and assume that the Z, are independent and that Zm} converges in law. It is shown that {X,/n} converges in law {n-‘z;,, to a certain infinitely divisible (i.d.) law. The following result allows alternative norming constants for {Zz_, I,,,}, Its proof is similar to that of Theorem 9 below. THEOREM

8.

,... }b e a sequence of constants satisfying

Letcr=l,y<~and{b,,,n=1,2 (9 b, >O,

(ii) b,,/n+m, (iii) b$,I/b,+xl/“(O<

x,6 < l),

(iv) b; ‘Z”, _ ,Z,,, s I, where Z necessarily has an i.d. law. Then

Remarks.

(1) A theorem of Bojanic and Seneta [3] shows that the form of the limit in (iii) is the only possible one and that (iii) implies b,,-n’/8L(n), where L(e) is slowly varying. (2) If E(sh)= h(s), l-h(s)=(l-s)*L and p = 03, then the norming varying and satisfies

(0<6< constants

b,, = n’/‘M(n),

1) where M(a) is slowly

M-S(n)L(n’/‘h4(n))+l,

yield the limiting

stable law E(e-@I)=

e-e”,

Seneta [19] stated such a result where L(x)= 4.

LIMIT

THEOREMS

We now consider

FOR THE TOTAL

1 and 8 < 1 PROGENY

the total progeny up to the nth generation: y,=x,+*..

+x,

A. G. PAKES

86 Clearly (cf. [lo]) II-1

where g,(s)=s and g,,+r(s)=sf(g,,(s)) (n=O,l,...) and h,(s)=E(s’m). When the {I,} are identically distributed, a< 1 and /3 < cc, then Yn/n+j3/ (1 - a) as.; see [lo]. With time varying immigration we have THEOREMP Let a < 1 and {b,} satisfv conditions (i), (iii) and (iv) of Theorem 8. Then

The results in [ 1 l] imply that

Proof: Let 0, = exp( - 0/b,).

&f(4) = l-

cle_

b

a)

+

0(bne2)+ r,(4),

”

where r,(0,,)= O(om) (m+co; uniformly in n). These may be used as in the proof of Theorem 7 to establish the result. Remark (2) following Theorem 8 applies here also, but with an obvious modification. Let now o= 1 and y < cc. When the I, are identically distributed and p< cc, it has been shown in [14] that m/n22

Y, where

An extension has been given in [ 151 for the time varying when b, = n. We now prove the following result.

immigration

case

THEOREMIO. Let a = 1, y < cc and {b,} satisfy the conditions in Theorem 8. Then with a, = nb,,, Y,,/a, s

Y, where m (1- e-‘“)Q(dx) x

87

GALTON-WATSON PROCESSES

and P(a) is the spectral measure in the canonical form of the L.S.T. of I [4, p. 4501. Under the conditions Remark. may take b, = n ‘/‘M(n) to get

of Remark (2) following Theorem

E(e-‘*)=exp

(

-l+s

8, we

88 1 .

The proof requires the following lemma. LEMMA If j, v

3.

are positive integers and j d v, then

II

de) = .J& j/v<

=exp{

-im[

k/n<(j+

h,,_k(e-e/bn) 1)/v

(exp( -0x(1--~)“‘)

-exp(

-0x(1--a)“‘)](

F)}.(8)

ProoJ: Denote the product by &(8). It is the L.S.T. of b;‘(SI,,l *.* +I,,, a=l-j/v and b=l-(j+l)/v. - Sr +t,,b]), where s,=r,+ Independence then yields

=44@)E(exp[-W l+In6l/bn)b;:InblSI+Inbll), and the result now follows using the canonical form of the L.S.T. of I, namely exp[ - J c(l - exp( - Bx))P(dx)/x] and (iii) of Theorem 8. The proof of the theorem follows those of the main result in [6] and Theorem 5 in [15]. First observe that (m+l)(l-s)-&(1-s)*<

1-g,(s)<(m+l)(l-s)

(O
1;

m= 1,2,...),

(9)

G,(0) = liyrstp

G(“)(e) < linm+stp

[~(I-41 n n j-1 j/vck/n<(j+l)/v

~“-ki&n,/“1(6)n))

where 0 < E< 1 and v is a positive integer. Using (9) we obtain

G,(e) < limsup fyF’Fh.,(exp( n-cc where p,, = O(RI,,j,,l(a,)

-$(I+[

= O(n’/bi)= G,(V
;]+p”))),

(10)

o(b; ‘). Lemma 3 thus yields [~(I-41 X +(eiiv)

-

j-l

(

. 1

The term in square brackets on the right of (8) equals

exp[ -0x(1--%)“‘I[

as v+cc, G,(B)<

(l-:,‘i’-l(J$)][l+o(l)]

and so, since E is arbitrary,

'/e-'e-exYcl-Y)""p(d,)aL

G(O)=exp

Similarly it can be shown that G(0)< The theorem now follows. 5.

THE FOSTER-PAKES

we finally obtain

liminfG(“)(e)

I.

and that G(O+)=

1.

MODEL

In this section we modify the immigration component so that immigrants may enter the population at time n + 1 only if the population size at n, U,,, is zero. This model, first investigated in [5] and [12], behaves in a qualitatively similar way to the model with unrestricted immigration when cr+ 1. When (Y= 1, p,S< cc, Foster [S] has shown that {logU,/logn} converges in law to the uniform distribution on (0,l). When p = cc we have the following result. THEOREM

II.

Let cu=l, y
and

1-h(s)=(l-&L(k)

(O
GALTON-WATSON

89

PROCESSES

where L(a) is slowly varying at infinity.

If 0 < S < i, assume in addition that

the sequence h,,h(f,_,)h(f,_,) (n=2,3 ,...) is decreasing. Then { U,/yn} converges in law to the distribution whose L.S. T. is ~(0)=1-C~‘(8-‘+x)-s(1-x)~-‘dx, 0

where C=(sins6)/n. Remark. Foster [5] has stated this theorem without proof for the special case where h(s) = 1 - (1 - s)~. ProoJ: We first observe that Tn= P( U,=O\ Uo= i)-C/n’-“L(n)

(n+co).

The proof of this is just as for Foster’s proof [5] when p < cc except that the renewal theorem of Garsia and Lamperti [9] is invoked when f < S < 1, and that of Williamson [22] when 0< S g f. We now use the expression

see [5] or [12]. Setting 0,=1-6/n (0<8‘+l. version of Spitzer’s comparison lemma yields upper and lower bounds fk(O,) of the form 1-(n/B+pk)-r, where p=y?e. Let Y be a positive integer and consider

v-l

=c

c

[ 1-h(l-(n/B+p(n-k))-‘)]T,.

A for

(11)

j=Ojn/vck<(j+I)n/v

When j=O,

the inner sum is dominated

above by

T,=(l-h(l-(n/B+p(l-vl)-‘)))

2

rk.

O
The first term on the right -Kn-‘L(n), where K=(8-‘+ p(l - v-‘))-~ and the sum-(C/S)(n/v)“/L(n). Hence limsup,,, T,,= O(V-“)-+O (v-+00). Clearly for j a 1, Tk may be replaced in (11) by the strictly decreasing sequence k&r l-k-’ Pk=cl-h(l-l/k)

=‘L(k)’

A. G. PAKES

90 A typical inner sum of (11) is dominated

by a term

= &Cv-'(j/v)s-l, where Kj=(O-‘+y(l-(j+ is arbitrary)

1)/v))-‘.

Now letting v+co,

lim sup S, < C ‘?(B n-+oa I0

-I+

y(l-

we obtain

X))_“dx

The right hand side is also a lower bound of liminf,,,S,, follows. The limit law of Theorem

(since E

and the theorem

10 has a density given by the mixture

(O
0

where 1)

,-q,fy/*_

%,y(9= q(y)=

- tlog(1 -y)

'

-cy-~-‘(1-y)6-Llog(l-y)

(O
and B6_,,6(x)=cx-8(1-x)8-’

(O
a beta density. The density a,(.) can be obtained by randomizing the beta densities Br_6,6(y) (r= 1,2,. . .) with the discrete distribution {q?}, where r-8

4, =

i Equation

6

(12) follows by observing 1

G(O)=

(r=1,2,...). i that

1 1 &-1,6(X)

0

and that the numerator the density

- l-8 xry6)

(1+ex)“-(xe)6

(i+exf

in the square brackets

(-xt >s-2(l_e-‘,x)

1 dx

is the Laplace transform

(t > Oh

of

(13)

GALTON-WATSON

91

PROCESSES

and the denominator can be recognized as that of a gamma density which when convolved with (13) gives the inner integral of (12). The case where p=ce and l-h(s)=(l-s)L((l-s)-‘) seems to be more complicated. The form of the limit distribution can depend on L(.). Foster’s argument shows that

where I(x)= J”; L(y)/ydy is slowly varying. If L(y)-_P(logy)” (O

P(g
(O
The case S=O is just Foster’s theorem. If, however, then the left of (14) converges to x (0 < x < 1).

L(y)-_P(loglogy)“,

I have pleasure in thanking Dr. E. Seneta for giving me access to [ 251 prior to its publication. It inspired Theorem 2, Lemma 1 and those parts of Theorems 3 and 7 not requiring Condition A. REFERENCES 1 2

K. B. Athreya and P. Ney, Branching Processes, Springer, Berlin, 1972. R. Bojanic and E. Seneta, Slowly varying functions and asymptotic relations,

3

Anal. Appl. 34, 303-315 (1971). R. Bojanic and E. Seneta, A unified

4 5

theory

of regularly

varying

sequences,

J. Math. Mufh. Z.

134, 91-106 (1973). W. Feller, An Introduction to Probability Theory and its Applications, Vol. 2, 2nd ed., Wiley, New York, 1971. J. Foster, A limit theorem for a branching process with state-dependent immigration, Ann. Math. Star. 42, 1773-1776 (1971).

6

J. Foster and J. A. Williamson, Limit theorems for the Galton-Watson time dependent immigration, Z. Wahrs. 20, 227-235 (1971).

7

C. R. Heathcote, Corrections and comments on the paper “A branching process allowing immigration”, J. Roy. Statist. Sot. 2I?B, 213-217 (1966). A. Garsia and J. Lamperti, A discrete renewal theorem with infinite mean, Comment. Math. Helu. 37, 221-234 (1961-1962).

9 10 11 12 13

process

with

A. G. Pakes, Branching processes with immigration, J. Appl. Prob. 8 32-42 (1971). A. G. Pakes, Some limit theorems for the total progeny of a branching process, Ado. Appl. Prob. 3, 176-192 (1971). A. G. Pakes, A branching process with a state dependent immigration component, Ado. Appl. Prob. 30, 301-314 (1971). A. G. Pakes, On the critical Galton-Watson process with immigration, J. Awl. Mufh. Sot. 12, 476482 (1971).

92

A. G. PAKES

14

A. G. Pakes, Further results on the critical J. Aust. Math. Sot. 13, 277-290 (1972).

15

A. G. Pakes, Limit theorems for the integrals of some branching processes, to be published (1975). M. Pinsky, Limit theorems for continuous state branching processes with immigra-

16

18

tion, Bull. E. Seneta, A remark E. Seneta,

19

142 (1969). E. Seneta, An explicit

17

Galton-Watson

process

with immigration,

Amer. Math. Sot. 78, 242-244 (1972). The stationary distribution of a branching process allowing immigration: on the critical case, J. Roy. Statist. Sot. 3OB, 176179. Functional equations and the Galton-Watson process, Ado. Appl. Prob. 1, limit

theorem

for the critical

Galton-Watson

process

21

immigration, J. Roy. Statist. Sot. 32B, 149-152 (1970). E. Seneta, The simple branching process with infinite mean, I, J. Appl. Prob. 206-212 (1973). R. S. Slack, A branching process with mean one and possibly infinite variance,

22

Wahrs. 9, 139-145 (1968). J. A. Williamson, Random

Walks

23

(1968). A. M. Zubkov,

of a branching

24

Appl. 17, 174-183 (1972). J. Galambos and E. Seneta,

20

25 26

Life-periods

and

Regularly

Riesz kernels, process

varying

Pacific J. Math.

with immigration,

sequences,

with 10, Z.

25, 393-415 Theory. Prob.

Proc. Am. Math.

Sot. 41,

110-l 16 (1973). E. Seneta, Regularly varying functions in the theory of simple branching processes, Ado. Appl. Prob 6,408420 (1974). E. Seneta, On invariant measures for simple branching processes, J. Appl. Prob. 8, 43-51 (1971).