asymptotic results for the extinction time of markov...

Asymptotic Results for the Extinction Time of Markov Branching Processes AllowingEmigration, I. Random Walk DecrementsAuthor(s): Anthony G. PakesSource: Advances in Applied Probability, Vol. 21, No. 2 (Jun., 1989), pp. 243-269Published by: Applied Probability TrustStable URL: http://www.jstor.org/stable/1427159Accessed: 04/06/2010 21:05

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Adv. Appl. Prob. 21, 243-269 (1989) Printed in N. Ireland

? Applied Probability Trust 1989

ASYMPTOTIC RESULTS FOR THE EXTINCTION TIME OF MARKOV BRANCHING PROCESSES ALLOWING EMIGRATION, I. RANDOM WALK DECREMENTS

ANTHONY G. PAKES,* University of Western Australia

Abstract The mathematical model is a Markov branching process which is subjected to

catastrophes or large-scale emigration. Catastrophes reduce the population size by independent and identically distributed decrements, and two mechanisms for generating catastrophe epochs are given separate consideration. These are that catastrophes occur at a rate proportional to population size, and as an independent Poisson process.

The paper studies some properties of the time to extinction of the modified process in those cases where extinction occurs almost surely. Particular attention is given to limit theorems and the behaviour of the expected extinction time as the initial population size grows. These properties are contrasted with known properties for the case when there is no catastrophe component. LINEAR BIRTH AND DEATH PROCESS; CATASTROPHES AND EMIGRATION; LIMIT THEOREMS

1. Introduction

Let (Z,: t--

0) denote the Markov branching process (MBP) with per capita jump rate a~ and offspring distribution {pj} where Po > 0 and pj > 0 for some j 1 2. As is well known, Zt is the size of a population in which individuals have independent lifetimes with an exponential distribution, mean a-'1, and just before their death individuals spawn j offspring with probability pj. It is the convention to set p, = 0. The basic branching property postulates the independent development of all individuals and this entails the independent evolution of extant family lines. Consequently, if T = inf {t

- 0:Z, = 0} is the time to extinction, and Zo = i > 0, then

T = max1ij•i Tj where Tj is the extinction time of the line of descent produced by the

jth ancestor and the Tj are independent and identically distributed (i.i.d.). In particular if F(t) = PI(T - t), where Pi(- ) = P(- I Zo = i), then

(1.1) P.(T - t) = (F(t))'.

Received 2 September 1987; revision received 24 March 1988. Postal address: Department of Mathematics, University of Western Australia, Nedlands, WA 6009,

Australia. This research was partially supported by N.S.F. Grant DMS-8501763 during a visit to Colorado State

University.

243

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244 ANTHONY G. PAKES

Moreover F(t) can be determined by quadrature as follows. Let f(s) =

ja-oP.si. Then F(t)

(1.2) J ds/(f(s) - s) - s) t.

In the non-supercritical case, m = poj 5 1, we have Pi(T

< oo) = 1 and it is easy to use (1.1) and (1.2) to obtain a number of results, reviewed in the next section, which in essence are properties of maxima of i.i.d. random variables. Briefly, these

properties pertain to the asymptotic behaviour as i--->oo of the quantities T, hi = E( T) and

(1.3) ri = lim Pi(T

> t)/PI(T > t). --*oo

Our aim in this paper is to explore the extent to which these properties change for certain modifications of the basic MBP structure which destroys the branching property. The modification we consider is the inclusion of a catastrophe or

large-scale emigration component. Denote the modified process by (X,: t _0). It is

defined to be the Feller process with state space N+ corresponding to the generator

iapj-i+l + K(i)dijI(o,i](j) (j 1, j i)

qij= jcpol{1>(i) + K(i)dioIN(i) (j = 0)

L-ia - K(i)(1 - dii)IN(i) (j = i)

where (a, {p,}) is as above, di, and K(i) -O,

Eo=0 di = 1 and I. (-)

is an indicator function. Thus when Xt = i the process can grow or decline by the reproduction of a

single individual during (t, t + h), or during this interval there is a catastrophe with

probability K(i)h + o(h) which reduces the population size to j with decrement

probability di,. Usually we set dii = 0 because catastrophes or emigrations always reduce the population size. Adequate motivation for the study of such a process can be found in Ewens et al. (1987) and Hanson and Tuckwell (1978), (1981).

Some aspects of special cases of this fairly general model have been reported in the literature. Almost all previous work distinguishes two special cases of the catastrophe rate, viz., K(i) = Ki and K(i) = KIN(i) where K is a positive constant. We will refer to these cases as Type 1 and Type 2 catastrophes, respectively. The likelihood of Type 1 catastrophes increases with population size, but Type 2

catastrophes can be regarded as being generated by an independent Poisson process. Most previous work has been concerned with particular offspring and/or

decrement distributions. Thus Kaplan et al. (1975) considered Type 2 catastrophes with a binomial decrement distribution. Brockwell et al. (1982) considered the linear birth and death process with Type 1 catastrophes and three particular decrement distributions, viz., binomial, uniform on {0, - --, i} for each i and the geometric defined by

d, = pqi--'(j= 1, .

, i - 1), = q'-l(j= 0)

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Extinction time of Markov branching processes 245

where 0 < q < 1. Later, Brockwell (1985) generalised the geometric to what we shall call the random walk case, defined by requiring the probability of a given population decrement to be independent of the current size of the population. More specifically, let {bj} be a distribution supported in N. In this paper we set

6i-j (1-5 j_5-i - 1,

i_- 1),

dij= { 6k (j = 0,i ).

Brockwell (1985) and, independently, Ezhov and Reshetnyak (1983) considered some properties of the extinction time of {XJ}, which we again denote by T, in the case of the linear birth and death process with Type 1 catastrophes and a random walk decrement distribution. We use BDPC1 to denote this process. The Markov branching generalisation is denoted by MBPC1. Discussion of the latter process is simplified once it is recognised that it is a random walk moving on N, which stops upon hitting {0} and jumps in continuous time at an instantaneous rate proportional to its current position. More precisely let p = a + K and

ai = (ar/p)pj+1_-1,=)(j) + (K/p)6_JI(-=olJ(j) (jE ),

then the generator of the MBPC1 is given by

qij = ipaj_i, (i, j > 1, i *j),

(1.4) qio= ip ak, qii = -ip and qj=O0, j-O0. ks -i

More generally, we may let {ai} be any distribution on Z. This approach was introduced in Pakes (1986). When the random walk with increment distribution {aj} is right continuous we recover the BDPCI; see Pakes (1987).

At this point we digress to comment on the notation used in the three above papers on the BDPC1. Pakes (1987) begins with (1.4) and sets aj = 0 if j _1.

He then finds it more convenient to use bj = a_. Ezhov and Reshetnyak (1983) use

oxi = pax-_ (j>- 0, *1). Brockwell (1985) denotes the birth and death parameters by A and M, respectively, whence p = A + / + K and

a, = A/lp, ao = K6o/p, a,-1= (M + Kd)/6 and aj= Kb6_/6 if j- -2.

Let D = C jai exist and q, = Pi(T < oo). Clearly (Pakes (1986)) qi = 1 iff D 5 0, a result proved analytically for the BDPC1 by Brockwell (1985). When D <0 he also proves for the BDPC1 that hi = E,(T) < oo and

hi "

(-pD)-' log i (i---> o0) and this result is extended to the MBPC1 under a second-order moment condition by Pakes (1986). This paper is the first attempt to derive properties other than those

pertaining to T. It presents a fairly complete investigation of the population size of the MBPC1 in the special case where

{ai} has a geometric left tail. This

simplification allows the use of familiar transform methods. Pakes (1987) gives a


fairly complete theory of the population size of the BDPC1 by relating its transition probabilities to those of a dual MBP. Further details are reported in Pakes (1988a) and Pakes and Pollett (1987).

In Section 3 we investigate further properties of T for these two special cases of the MBPC1. For the MBP it is known (see (2.4) below) that T - hi converges in law as i-->oo to a double exponential distribution. We obtain similar results (Theorems 3.2.1 and 3.2.2) for the above special cases but the limiting distributions are complicated modifications of the double exponential.

When D =0 and V = E ajj2 <00 then in both cases a normed version of T converges in law to the extreme value distribution exp(-1/x) (see (3.3.1) and Theorem 3.3.2). These results coincide in form with one for the critical MBP ((2.8) and (2.9) below) and hence they must be special cases of a yet unproved general theorem for the MBPC1. When V = o0 limit theorems exist under a regular variation condition and they differ in form for the two cases. The limiting distribution for the MBPC1 with geometric left tail is the same extreme value law as for the MBP, see Theorem 3.3.1 and (2.8). We also obtain the asymptotic behaviour of r, and this may be compared with the known exact value for the MBP, viz., r, = i.

In Section 4 we investigate a special case of the MBP with Type 2 random walk catastrophes (denoted by MBPC2), the BDPC2. The simplifying random walk structure of the MBPC1 is no longer present and consequently the theory of the MBPC2 is less well developed. When m

_ 1 it is clear that q, 1, but the situation is

less clear when m > 1. Pakes (1988b) has shown that qj 1 if E 6, log j = oo and that this condition is necessary in the case of the BDPC2. Thus in Section 4 we restrict our discussion to the BDPC2 in the cases where it becomes extinct almost surely. An account of the population size of this process in all cases will be given in Pakes (1988c).

In the subcritical case the catastrophe component is manifested by a reduction in the second-order term of hi as given by (2.6)--see Theorem 4.2.1. As for Type 1 catastrophes, T - hi converges in law to a modified version of the double exponential distribution. In the critical case h, < o0 and under a regular variation condition T/h, converges in law (Theorems 4.3.1 and 4.3.2).

The most interesting case is the supercrvtical one which bears many similarities to the subcritical MBP with an immigration component so intense that no limiting distribution exists (Pakes (1979a)). In the present case hi can be infinite or finite (Theorem 4.4.1) and there is a limit theorem for T which is similar to known results for the population size of branching processes with intense immigration (Barbour anakes (1979)). This is discussed in Theorem 4.4.2, whose proof borrows from the technique of the last reference. Additional hypotheses on the decrement distribution yield corollaries asserting convergence in law of T/h,.

The reader is cautioned that one item of notation introduced above will be

changed. In Sections 2 and 4 we always use p to denote the per capita branching


rate, i.e. a is replaced by p. This will facilitate comparison of the results in Sections 2-4.

2. The Markov branching process

When m - 1 it is clear from (1.1) that

(2.1) Pi(T

> t) - i(1- F(t)) (t- oo) and hence that

(2.2) ri = i.

The tail behaviour of F(t) can be obtained from (1.2) under various conditions. For example let m < 1 and

/[ (1 1-ms)

] c=exp 1-s f(s) -s >0,

which condition is equivalent to

> Pjj log j < .

Some manipulation with (1.2) (cf. Zolotarev (1957)) yields

(2.3) e'(1 - F(t))---> c

where v = p(l - m). It follows that

1 - F(t + x/) = e-x

t--,= 1 - F(t)

and hence that F(-) lies in the domain of attraction of the double exponential distribution (Leadbetter et al. (1983), Theorem 1.6.2) and consequently (Leadbetter et al. (1983), Corollary 1.6.3), for x e R,

(2.4) lim Pi(vT -5x - log (ic)) = exp (-e-x).

t--*oo

Thus in a certain sense T - v-1 log i when i is large. The last comment has a more precise sense in terms of expectations. Integrating

the tail version of (1.1) and making a substitution using the differentiated version of (1.2) yields the well-known representation

(2.5) hi = p- 1 1 i

ds f(s)-s

and hence

(2.6) hi = v-1(log (ic) + y + o(1)) (i-- oo)

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where y is Euler's constant. In particular

(2.7) hi - v-1 log i.

Now suppose m = 1 and

f(s) - s = (1 - s)1+ 'L((1 - s)-1)

where 0 < 6 - 1 and L( ) and is slowly varying (SV) at infinity. If V =f"(1-) < oo then L(x)-+ V/2 as x ---> oo. Let

(s) = p-1 dx

0f(x) -x" It follows from (1.2) that ds(s) and 1 - F(t) are inverses of each other, d (. ) is

regularly varying (RV) at the origin with index -6-1. Consequently F(t) lies in the domain of attraction of a Type-II extreme value distribution (Leadbetter et al.

(1983), Theorem 1.6.2) and that, for x > 0,

(2.8) lim Pi(T

5 xd(i-1)) = exp (-x-V6). i--00

If V < oo then

(2.9) d(i-) ~- 2i/pV (i --oo).

Under this condition hi = oo, but if 6 < 1 it is not hard to show that

hi - (i-1)F(1 - 6) (i-- -oo). We take the principal consequences of (1.1) and (1.2) to be (2.2) and (2.6)-(2.8).

3. The MBPC1

3.1. Introduction. Suppose - <D <0, called the subcritical case, let b =-D and v = p |D1. For the MBP D = 1 - m. Pakes (1986) generalised (2.6) by showing that

(3.1.1) hi - v-1 log i (i-> oo).

In the following subsections we see that T - v-1 log i has a limiting distribution in our two special cases, Cases I and II respectively, corresponding to the MBPC1 with a geometric left tail and the BDPCI.

3.2. The subcritical case. We begin by considering Case I where we have

aj = ppq-;-1 (j : -1) where 0 < f, q < 1 and p + q = 1. Let

A (s ) = > a s',

and

a(s)= A

(s) + tip 1 s--q


which has a single zero q in (0, 1). Define

I(s, 6) = exp -(O/p) A(y) dy

and

Ji(O) = yi-'l(y, 0) dy.

Pakes (1986) has shown that there is a number E (0, v) such that J,(O) is holomorphic in re 0 > - v, and

ri = lim P,(T > t)/PI(T > t) (3.2.1) .oo

Monotone convergence shows that Ji(- )-->0 as i-- oo and hence the constants r, can never be asymptotically linear, cf. (2.2). The precise rate of decrease of Ji(- () will be derived as part of the proof of a limit theorem for T.

To formulate this define

K(O) = q101 A(y)I(y, 6) dy.

yo y-q

This can be put into the form

K(O) =

p(1 + 0

foe-'tA(t) dt)

where A( ) is an integrable, continuous and non-negative function whose Laplace transform is holomorphic in re 0 > - v. The above constant ? is the unique solution in (0, v) of the equation K(-6)= 0. Corresponding to the MBP let

c=exp[-f (

S- DA(s)) ds+log(1 -q)]

We recover the MBP case by setting q = 0, whence f(s) = fl + sA(s). We prove the following extension of (2.4).

Theorem 3.2.1. Let D > 0 and Ej2_x

ajj logj < o0. Then

lim P,(vT -x + log (ic)) = F(x) i---*o

where II _ =p F(1 + i6)

70e_ F(dx) -pF( + 0) f.K(vO)

Remark. The gamma function in the limiting Laplace-Stieltjes transform (LST) is the LST of the standard double exponential distribution. Thus F(x) can be regarded as a modulated version of this distribution where the modulating function depends in


a complicated way on the increment distribution. The reciprocal of K(O) is not an LST and it does not seem possible to determine K(O) in any particular situation.

Proof. First note that the logarithmic moment condition is equivalent to c > 0. It was shown by Pakes (1986) that

(3.2.2) hi(O) = Ei(e- ") = ipJi(t)/K(6)

and that h;(O) is holomorphic in re 0 > - . Since K(O) = p, a continuity theorem for the bilateral LST (Pakes (1974), p. 23) shows that it suffices to prove for 0 > -1 that

(3.2.3) lim i0+1J,(vtO) = c- V(1 + 0). i--*-o

To see this let

y(s) = exp f(1lY'-DA(y))dy-log(1-q)]

and observe that I(s, v6) = [(1 - s)y(s)]0. Let 0 < E < 1. Since 0 is fixed and i is

large-we have for i0> -1,

iJ.(Ov) = 0(1 - E)' + i yi-l(1 -y)0(y(y))0dy 1-E

= 0(1 - E)i + i(c- + r(E)) yi-l(1 - y) dy

= 0(1 - E)i + i(C- + r(E))B(i, 1 + 0),

where r(E) = O[c-1 - y(1 - e)]. Now, limi,= iF(i + 1)/F(i + 0 + 1) = 1 (Abram- owitz and Stegun (1964), 6.1.47) and (3.2.3) follows by letting i-- o0 and then E-+ 0.

The next result follows from (3.2.1) and (3.2.3) with 0 = -~/v.

Corollary 3.2.1. As i-+ oo,

ri - (ic) /T(1 - /v)/J(-).

Since V/v < 1 we see that the presence of catastrophes gives a stochastically smaller value of T for large values of X0 in comparison to moderate X0 values than occurs without catastrophes.

We consider the BDPC1 defined by setting aj = 0 if j > 1 but a1 > 0 and aj > 0 for some j<-l. It is convenient to set b;=a_a

(j-_-l), f(s)= jiobb_,s' and

b(s)=f(s)-s. Let (Y,:t'

0) denote the MBP with per capita jump rate p and offspring probability generating function (p.g.f.) f(s). If

Ti(t) = Pi(T

> t)


and F(s, t) = E(s, I Yo = 1) then

(3.2.4) r

ii(t)s'

= ( b(F(s, t))

i1 b(s) 1 - F(s, t)' see Pakes (1987), (2.10).

The subcritical case corresponds to m = f'(1)> 1. In this case b(- ) has a single zero q in (0, 1). Let q <S< 1 and define

B(s) = -p-1 dy/b(y).

This function is strictly increasing on (q, 1), B(q+)= -oo, B(s) = 0 and B(1-)= oo It follows that B( -) has an inverse function B(x) defined and strictly increasing on R, A(-oo) = q, B(0) = J and B(oo) = 1. The integrated backward equation for (Y,) shows that if s > q then

(3.2.5) F(s, t) = B(B(s) - t). Let D = m- 1, v = pD and

c = exp{f(1 +4 )dss+log(1 -q). e (1 (s)

We now prove the following limit theorem.

Theorem 3.2.2. When D < 0 and Ejax a_jj log j < oo there is a probability measure

•(u ) supported in R+ and defined by

-#o0e

-(dx) -b(B(v-1 log -1))

of -•J=D(1 - A(v-1 log 0-1)).

As i -> o,

(3.2.6) Pi(vT < x + log (ic))-- fxo) (dy).

Proof. We start by defining a signed measure supported on N+ and attributing mass

@i(t)= -i(t)

- 1i_l(t)

to {i} where we define ro(t) = 0 if t > O0. Let P(s, t) be the generating function of this measure, whence

(3.2.7) P(e-j,

t) = f -

ex~dxP[x(T > t).

It follows from (3.2.4) and (3.2.5) that for s > q

(3.2.8) P(s, t) = .

)(1-B(B(s)-t)-

b (s) 1 - [B(B(s)- t)]

We let t 00 and choose s = s,- 1. It follows that the first factor on the right-hand side -/D-1


Define y(s) by -

+( -

dy _ log

(1_)

logy(s)= = b(y) 1-y

and s, = exp (- On,) where nt = ce -.

Obviously vB(s) = -log y(s) - log (1 - s) whence

(3.2.9) B(st) - t- v-' log 6.

It follows from (3.2.7)-(3.2.9) and the extended continuity theorem for LSTs (Feller (1971)) that

Plyn,•l(T > t) a ([0, y])

where p( ) is a signed measure. Since /(0+) = 1, p(- ) attributes unit mass to R+, as does the measure defined by {((( , t)} for each t. Setting i = [y/n,] and y = e-x we obtain

vt = log (ic) + x + o(1) (i -> oo)

and (3.2.6) follows. We can also deduce from (3.2.6) that p(.

) is a probability measure. Since b(q) = 0 we deduce that f(oo) = 0 and hence (({0}) = 0. It follows that the limiting DF in (3.2.6) is not defective, and the proof is complete.

It was first shown by Ezhov and Reshetnyak that ri exists and is defined by

(3.2.10) rs' =(sal/b(s))exp

-d dx/b(x)] (0 s <q)

where d = 1 -f'(q). An alternative approach by Pakes (1987) shows that ri = ixi where

{xi} is the v-invariant measure of (Yt). Let r = d/D and

*(s)= exp-d 1 1 1 ]dy. = exp -d b(y) d(q- y)+ L(1- y)

The following result gives the asymptotic behaviour of the ri.

Theorem 3.2.3. Under the conditions of Theorem 3.2.2, y* = y*(1) is finite and positive and as i -oo

r, _ iry*al(1 - q)/DF(1 + r).

Proof. The first assertion is obvious. It follows from (3.2.10) that

(3.2.11) a1 risi-'1 = [(q - s)/b(s)•y*(s)(1

- s)-r

The first factor is the generating function of a discrete aperiodic renewal sequence

{Vi,} and

vi--, v, = (1 - q)/D;


see Pakes ((1979b), pp. 290 and 292). Writing y*(s)= C yis' it is easily seen, since

Ej=o yj is positive and finite for all sufficiently large i, that

ai = V

tli-jyj I

"007*" j=O

It follows from (3.2.11) that

ri = a, oi- j=0 J

and the assertion readily follows because the binomial coefficient ~jr-'/F(r) as

j--> O0 Observe that the dominant component of the centring constants in Theorems

3.2.1 and 3.2.2 are the same and are what would be expected from (3.1.1). The centrings in these theorems also contain a constant term, log c, but c is defined differently in each case. Both the definition of c and the specification of the limiting distribution in Theorems 3.2.1 and 3.2.2 depend in a detailed manner upon the left-hand and right-hand tails, respectively, of the increment distribution. Presum- ably our results are special cases of a general theorem, but it is not yet clear what form this might take. Finally we remark that in contrast to Corollary 3.2.1, the constant r in Theorem 3.2.3 can take any positive value.

3.3. The critical case. We begin by considering Case I with D = 0. Our first result is very similar to (2.7). For 0<s < 1 let

1-S

(s) = p-1 A(y) dy,

and recall the notation introduced at the beginning of Section 3.2.

Theorem 3.3.1. Let A = A(1) and suppose that

A(s) = A - (fl/p)(1 - s) + (1 - s)'+6L((1 - s)-1)

where 0 < 6 1 and L is SV at infinity. Then, for x > 0

lim Pi(T

- x4(i-1)) = exp (-x-l'1). i-1-*o

If V = j2aj < 00 then

(3.3.1) lim Pi(pVT/2i

5 x) = exp (-x-1). i---*o

Proof. We replace 0 in hi(O), (3.2.2), by 6/4(i-1) which -->0 as i--oo. For any t > 0 there exist functions 0 < e(t) < t satisfying d(1 - u,(t)) = t (j = 1, 2). These are related to functions s,(t) (j = 1, 2) defined by Pakes (1986) via o,(t) = 1 - s,(t). It follows from Lemma 5.1 in Pakes (1986) that

I o(t) - p | = O(tl/2) (t-- 0),


1- ax(t) 0 exponentially fast as t T oo and u2(t) 0 as t T oo. Our hypotheses imply that

A(s) = [(1 - s)I"'LI((1 - s)-)]-x

where LI(x) - L(x) (x-oo) if 6 < 1 or if 6 = 1 and L(x)-- oo, otherwise Lx(x)- L(oo) + l/p2 = V/2 < oo. It follows that

d(s) - (pbs'+6L2(1as))-1 (S -> 0)

where L2(x)" L1(x) and we conclude that u2(t) is - -'-varying at infinity (Seneta (1976), p. 24).

It was shown by Pakes (1986) that

( pt) (1-

ax(t))

2(t) A(t) =

q\(1-1

,) + 04

a(t) - p p - 02()

whence A(O) is SV at the origin and we conclude that K(O+) = p. It can be inferred from (5.12) in Pakes (1986) that

iJ(6) = if (1 - ax(t))'-'a'(t)e-

dt

-

ifo

(1 - o2(t))i-~ (t)e- dt

and the first term on the right is O(q')-+ 0 as i oo. By making the change of variables t = ysi(i-') we obtain

iJi(U /(i-1)) = f

e- dF(y) + o(1)

where

Fi(y) = [1 - 02y -1)'i

~ [1 - y-1/6o2(4(i -))]i-> exp (-y -), and we have used the regular variation of 02( ). The theorem now follows from

(3.2.2) and the continuity theorem for the LST. In the present case it is not clear whether ri exists. Pakes (1986) obtained the

following weak result for Case I. When E Ij3 ai < o0 there exists a sequence t,,--m such that

P (T > t,)/PI(T > t,)-- ip + q.

The MBP case corresponds to p = 1 and q = 0. Now consider Case II with D = O. Here b(s) > 0 if s <1 and hence

p(s) = p-1 dy/b(y)

is strictly decreasing in (0, 1) and @(0+) = oo. Let ( . ) be the inverse of 2( ) and


b*(s) = b(1 - s). The reasoning leading to (3.2.7) and (3.2.8) yields

(. ?s(1 - s) b*(I!(t + V(1 - s))) (3.3.2)dP(T > t) b*(1 - s) " (t + V(1 - s))

where s = e- .

If 0 < 6 -

1 then (1 + 6")-1 is the LST of a DF F(x) which has the representation

F(x)= fHa(x, t)e-tdt

where Ha( , t) is the DF of the stable subordinator whose Laplace exponent is 06. We now prove the following result.

Theorem 3.3.2. Suppose b*(s) =sX+L(s-') where L( ) is SV at infinity and 0 < 6 1. Then

lim P,(T 5 xO(1/i)) = 1 - F(x-X/). i-.-*o)

If V < oo then (3.3.1) holds.

Proof. We replace 0 in (3.3.2) by Oi(t), whence

1 - s = iOO(t)(1 + o(1)) (t--> oo). As above,

4(s) = (p6s L1x(1/s))-1

where LI(x) - L(x) and L(x)-- V/2 when the latter is finite. It follows that A(t) is

-6-X-varying at infinity. The uniform convergence theorem for regularly varying functions (Seneta (1976), p. 2) yields, as t ,00

(1- s) = 04(o(t)(1 + o(1))) = -"t(1 + o(1)) and hence

A(t + 4(1 - s)) ~ (1 + l- ?)-x•(t).

It follows that the right-hand side of (3.3.2) converges to (1 + ")-1. We conclude from the extended continuity theorem for LSTs that

Pfy/1(ol(T > t)--> F(y).

Let y =x-x" and i= [y/l(t)]. Then t-->oo as i-->oo, i-(xX" (t))-1-i+1

and hence t - xV(1/i). The theorem follows.

Under an additional moment condition Ezhov and Reshetnyak have given a

partial proof of the existence of r1 and a representation as a convolution. Pakes

(1987) has shown without any extra assumptions that

ri = ali 1i

when { xi}

is the invariant measure of the (critical) dual MBP mentioned in Section


3.2 and given by

SnxisI'= dy/b (y).

Known results about the asymptotic behaviour of {(}i) transfer to {ri). In particular if V < oo then (Yang (1973))

ri - 2ali/V. If V = oo then ri = o(i).

When V = oo the limit distributions in Theorems 3.3.1 and 3.3.2 differ in type, but they coincide when V <oo and it then seems likely that for the critical MBPC1 we will have P,(pVT - 2ix) -- exp (- 1/x).

4. The BDPC2

4.1. Notation. The offspring distribution is given by Po = a and P2 = 1 - a = b > 0. Let si(s) =

Eijx 6s' be the p.g.f. of the decrement distribution, H(s) = (1 -

q(s))/(1 - s) and a = K/p. With 'r,(t) = P,(T > t) the backward equations yield the

generally applicable equation

- r(t) = -(K + pi)~i(t) + pi pj;Ti+-l(t)

+ K>

dji•_-j(t). at j=o j=o

Let r(s, t) = ri 1 "i(t)s' and ^ = T(s, 0) = f" e- &(s, t) dt. For the BDPC2 we obtain

(4.1.1) Or - s(1 - s)-1 = p'(as2 - s + b) + (pas - pb/s - K(1 - O(s)))Z

where V' = (a/as)T. We use this equation to obtain solutions for r(s, t). 4.2. The subcritical case. Assume a > b and set

r = b/a, v = p(a-b) and B= a/a.

Then (4.1.1) can be written as

(4.2.1) '- r -

S2 BH(s)0 1s (4.2. 1) r'-+ + - - =+-

S.

s(- s)(r- s) r - s v 1-s s-r ap(1-s)2(r -S)

In the region s > r a suitable integrating factor is

I(s, 0) = s-l(1 - s)1,-/v( _- r)1++/v exp (Bq9(s)) where

qg(s) = H(y) dy

wy-r

where r < z <1. Clearly qg(s)- -oo as s-+r+ whence

i(s, 6) = (s/pa)(1 - s)l+o/v(s - r)-l-"le-() (4.2.2) x (1 - y)-l-Ov(y - r)O/Iea(Y) dy.


By comparison with the subcritical MBP it is obvious that hi<oo. Since h(s) = Eia1 his' = i(s, 0) we obtain

h(s) = s(pa(1 - s)(s - r))-le-B'") (1 - y)-leB?(Y) dy.

Let q = q(1) which is finite iff E(log Y)? <o where Y has the decrement distribution. Assume this is valid. Then the above expression can be cast into the form

h(s) = s(pa(1 - s)(s - r))-f[-log (1 - s) + log (1 - r)

(4.2.3)

- exp (-B9q(s)) (exp (Bqp(s)) - exp (Bqp(y)))/(1

- y) dy .

We now prove the following result.

Theorem 4.2.1. The quantity

M = f(1- exp (-B(q0 - ( )( y)))/(1 - y) dy <

iff E(log Y)2 < 0. When this condition is satisfied,

hi = v-l[Ilog i + y + log (1 - r) - M + o(1)l (i --> oo)

where y is Euler's constant.

Remark. This expansion can be compared with (2.6). For the subcritical birth-death process we have c = 1 - r. In the absence of emigration B = 0 and hence M = 0 also. We expect emigration to reduce the expected extinction time and we see that this is manifested by a reduction of the second-order term in the asymptotic expansion of hi.

Proof. The integrand in (4.2.2) increas~es with s and the limit as s -- 1- of the integral is finite iff

S(q --

p(y))/(1 - y) dy <d .

But this integral is

1( 1 -1x)

= - 6i

xlog (1- x) dx. - -LVi,1 -x


As i--> oo, the integral in the last term is

1 j-1 j-i 0

-f1xk log (1 - x) dx = E1 (i(i + k))-1 k=O k=O i=1

j-1 k

=- > k1 k- i-+ (1) x- logx dx = -(log j)2 k=1 i=1

and the first assertion follows.

It follows from (4.2.3) that

h(s) = (-log (1 - s) + log (1 - r) - M + o(1))/v(1 - s)

and the second assertion can be deduced from this. To invert the transform fr(s, 6), let

r(l - s) + (s - r) exp (- vt) F(s, t)= 1 - s + (s - r) exp (-vt)

which is the p.g.f. of the population size of the supercritical birth-death process with jump rate p and offspring distribution given by Po = b and P2 = a.

Lemma 4.2.1.

= sv exp (- vt)

(s, t)) 1 - s + (s - r) exp (-vt)exp [-B(q(s) - q(F(s, t)))].

Proof. Let R(s) = (s - r)/(1 - s) which is increasing on (r, 1). The terms contain- ing 0 in (4.2.2) can be written as

(R(y)/R(s)) v= fo e-6(t

- v-' log (R(s)/R(y))) dt,

where 6( ) is the delta function, and hence

r(s, t) =' exp (B(s)) (1 - y)-16(t - v-1 log (R(s)/R(y)))e"'9Y) dy. pa(1 - s)(s - r) J,

The argument of the delta function is 0 iff

R(y) = R(s)e-"

which has the unique solution y = F(s, t). Making the substitution y = F(s, u) yields the assertion of the lemma.

Next we use Lemma 4.2.1 to obtain a limit theorem for T.

Theorem 4.2.2. Let a > b and E(log Y) < m. The quantity

(4.2.4) f() = (1 + )-l exp [-- ( + r)+a ep B -]


is the LST of a probability measure y( - ) and for x e R,

lim Pi{vT _-x

+ log (i(1 - r))} = ~((e-x, oo)). i--oo

Proof. We use the construction leading to (3.2.6) and set s, = exp (- On,) where

n, = (1 - r) exp (- vt).

Then 1 - st, 0(1 - r) exp (- vt) whence

F(s,, t)--- r + (1 - r)/(1 + 6) and

(1 - r) exp (- vt) > (1 + 6)-1

1 - s, + (s, - r) exp (-vt)

It follows that P(st, t)- (06). Clearly f(0+) = 1 and s(0) is the LST of a probability measure tI concentrated

within RI+ if the exponential term in (4.2.4) is completely monotone. To prove this it suffices to show that y'(0) is completely monotone, where

y(O) = = - ;p((l + rO)/(1 + 0));

see Feller ((1971), p. 441). But

y(0) = H(s)/(s - r) ds 1+re)1(1++)

whence

' = - (r + r)](1 -r)#,

'(a)= 1-• r+l+-

and ?@(r + (1 - r)/(1 - 0)) is the LST of a probability measure and hence so also is

0-1(1 - (1 + rO)/(1 + 6))). We conclude that p( - ) is the convolution of the standard exponential law and an

infinitely divisible law. Note that as -- 0,

and it follows that the infinitely divisible component is not compound Poisson--its Levy measure is infinite.

The extended continuity theorem for LSTs yields

PlyIn,1(T > t) M([0, y]) and the limit theorem follows as in the proof of Theorem 3.2.2.

When B = 0 then y((x, oo))= e-x and we recover (2.4) for the birth-death case. As a less trivial example, suppose 61= 1, i.e., the exogenous catastrophe component removes exactly one individual at each catastrophe event. Then H(s) 1 and


y(0) = log (1 + 6). Thus 1(0) = (1+ 6)-1-8 which corresponds to a gamma density. The limiting DF of T is

(r(1 + B))-1 eI yBe-Y dy.

Finally suppose the decrement distribution is geometric: 6b = (1 -p)pj-1 for j il 1 and

0-5p < 1. Then H(s) = (1 - ps)-1 and

12(0) = (1 + 6)-1(1 + 0(1 - rp)/(1- p))-B/(1-p)

whence y( ) is the law of the convolution of a standard exponential law and a two-parameter gamma law.

To ascertain the existence and asymptotic behaviour of the ri we must obtain an expression for r(s, t) valid for 0

- s < r. We approach this by solving (4.2.1) using

the integrating factor s-l(1 -s)l- /v(r - s)l+O/Vexp(Bqp*(s)) where qg*(s)=

~o H(y)/(r - y) dy. The steps leading to Lemma 4.2.1 yield, for 0 -

s < r,

sv exp (- vt) r(s, t) exp (- exp [B(qp*(s) - q9*(F(s, t)))]. pa(1-s) 1-s-(r-s)exp(-vt)

Now qp*(s) = H(r) log (r/(r - s)) - ?p(s)

where p9(s) = Jf (H(r) - H(y))/(r - y) dy, and some algebra yields

r(1 -s) Tp*(F(s, t)) - qg*(F(0, t))-- H(r) logr( ( (t-- o). r-s

We conclude from this that lim_,

r(s, t)/Tl(t) exists and hence from the continuity theorem for p.g.f.'s that {ri} exists. Its generating function is

Srisi = s(1 - s)-2-BH(r) exp (-Bp(s)).

If E(logY)<oo then p=0p(1-)<oo and if K=exp(-Brp) one may use a Tauberian theorem to deduce that

ri - (K/r(2 + BH(r)))il+BH(r)

The Tauberian theorem is applicable because (X,) is a stochastically monotone Markov process and hence the sequences {r1} and {hi} are non-decreasing-see Pakes (1988b).

Thus adding a Type 2 emigration component to the birth-death process increases the growth rate of the ri. This can be understood as follows. The pressure of emigration is the same no matter how large Xo = i is. Thus when i is large emigration is less important than the natural rate of decrease than is the case when i is small. Indeed as Theorem 4.2.2 shows, for large i, the leading component of T is independent of the emigration component. Thus lr(t) will be similar to the same


quantity without emigration, but Tl(t) will be rather smaller with emigration than without it.

4.3. The critical case. We assume a = b = ? in which case the integrating factor for (4.2.1) may be taken as s-1(1 -s)2exp[-Bq*(s) -2/p(l - s)] and the solution is, for 0

:_- s < 1,

(4.3.1) i(s, O)I(s, 0) = (2/p) (1 - y)-' exp [-Bq)*(y) - 20/p(1 - y)] dy.

First we prove the following result.

Lemma 4.3.1. When a = b, hi = Ei(T) < oo for all i and

h(s) = [2s/p(l - s)2J(s) f[(1 - y)J(y)]- dy

where J(s) = exp (Bqp*(s)).

Proof. Using the monotone convergence theorem and an extended continuity theorem for generating functions it suffices to show that f' ((1 - y)J(y))-' dy < 0.

First suppose that I = EY < oo. Then

qp*(s) = -is log (1 - s) -

S p - H(y) dy

and by setting z = (1 - y)-' in the second integral we obtain the representation

(4.3.2) (J(s))-I = (1 - s)B•L((1 - s)-1)

where

L(x) = exp [-B f(~- H(1 - 1/z))z-1 dz

is SV at oo. Thus the integral in question is indeed finite. This continues to be the case when p = oo because J-1(s) decreases to 0 faster even than above.

This result is interesting in view of the fact that for the birth-death process hi is 'just infinite'. By this we mean that for the critical MBP hi = oo if the offspring distribution has a finite variance, but h < oo if the offspring distribution has a slightly fatter upper tail-more exactly iff

ff1 S

ds <00. f(s) - s In the present context the lemma shows that Type 2 emigration entails h < 00.

When 61 = 1 it is easily seen that

hi = 2i/pB.

The following working shows that an asymptotic version of this relation is valid under the following regular variation condition.


Suppose that

(4.3.3) 1 - 2(1 - s) = sL(1/s)

where 0< 6 -1 and L( ) is SV at infinity. It follows that when 6 < 1,

I-1/x

R(x) = B H(y)/(1 - y) dy

= (B/(1 - 6))x1-6L1(X)

where L1(x) - L(x), and if 6 = 1 then

R(x) = B fy-L(y) dy

is SV at infinity and R(x) T' 0 as x ' oo. We then have h(s) = 2sQ(1 - s)/p(l -s)2 where

Q(s) = eR(l/s) z-le-R(z) dz.

Let y = R(1/z) and suppose first that 6 = 1. Then

(4.3.4)Q(s) = eR(s)(s) (BL(1/z))-e- dy

R(lls)

and if p = L(oo) < oo we conclude that Q(s)-- 1/Bp as s-+ 0. If p = oo we use

d - (L(1/z))- = -L-2(1/z)(z-1L'(1/z)/L(1/z)) dy

=o(L-2(1/z)),

where the last equality follows because L is a monotone and differentiable SV function (see Seneta (1976), p. 60). Integration by parts yields

(L(1/z))-te-Y dy = (L(l/z(x)))-le-x + f o(L-2(1/z))e- dy.

Since 1/z(R(s-l))= s- we conclude that in both cases

Q(s) - 1/(BL(1/s)) or h(s) - 2/pB(1 - s)2L(1/(1 - s)).

When 6 <1 (4.3.4) continues to hold in the form

Q(s) = e"R') (zR(1/z)/R'(1/z))e- dy/y. R(l/s)

Since R is a monotone and differentiable RV function with index 1 - 6 the z term in

the integrand --(1-

6)-1 (Seneta (1976)) and the asymptotic behaviour of the


exponential integral finally yields Q(s) - ((1 - 6)R(1/s))-1. Thus h(s)- 2/pB(1 -

s) +"L((1 - s)-1) and a Tauberian. theorem for power series finally gives

hi - 2i6/pBF(1 + 6)L(i) (i oo) whenever 0 < 6 1.

We now consider limit theorems for T and look first at the case where y <oo. Theorem 4.3.1. If a = b = ? and M < oo then, for v > 0,

lim Pi(pT/2i - v) =

(F(1 + Bts))-1 f pe- dy. i--0oo .1/

Remark. The limiting distribution is not an extreme value type when B > 0. When B = 0 we recover (2.8), but observe that when B > 0 we do not assume the finitude of V.

Proof. We can invert the transform (4.3.1) as in Section 4.2 to obtain

r(s, t) = (s/(1 - s))(1 + pt(1 - s)/2)-1 exp B(qp*(s) - qp*(F(s, t)))

where

F(s, t) = 1 - (1 - s)/(1 + pt(1 - s)/2). As above, we define P(s, t)=(1 - s),(s, t), so that (3.2.6) holds, and we set s = s = exp (-20/pt). Then

(1 - s,)/(1 - F(s,, t))-- 1 + 6 and

F(s,, t)

qg*(F(st, t)) -

qg*(s,) =

t)H(y)/(1 - y) dy

St

- ip log ((1 - st)/(1 - F(st, t)))--+ l log (1 + 6).

It follows that P(st, t)- (1 + 6)-1-B4 and hence, as t-- oc,

Pe[xl(T > t)-- (7(1 + Bs))-1 f yBe-y dy.

The assertion follows on setting i = [ptx/2] and x = 1/v.

Next we assume (4.3.3) is in force with q = oo. Let &1(i) = (1 - O(1 - 1/i))- and, referring to the notation introduced just before Theorem 3.3.2, let H6 (x) = H6 (x, 1).

Theorem 4.3.2. Let a = b = - and suppose (4.3.3) holds with I = oo. Then, for v >0,

limn P,(pBT/2(i) 5 v) = 1 - H(v-). i-.oo

Proof. For each t > 1 we choose n, so that

(1 /n,) = t


and set s, = exp (-On,). As t -- o we have nt-- 0 and (4.3.3) implies nt = t-1/6M(t) where M( ) is SV at infinity. When 6 = 1, M(t) --0 as t --oo and hence in all cases tn,-- 0. It follows that t(1 - s,)--0, 1 - F(st, t) 1 - s, and in addition t(1 -

Now

fF(s,t) 1-s H(y)/(l - y) dy = -F(,

t) z 6L(1/z) dz

whence

--st q*(F(st, t)) - q*(st) L((1 - st)-1) z-2-6 dz

[ 1 - F(s,, t)

1 -__ (s_) 1 - se V ]1 (1 + 6)(1 -

st) [1-

F(s,, t))

t - 9 (s,) - [(1 + pt(1 - st)/2)1+" - 1]-* p6P/2. (1 + 6)(1- st)

It follows that P(s,, t)- exp (-pBO /2)

whence

Pxi/n,l(T >

t)"- H6(x(2/pB)1/6).

Let i = [x/n,]. The uniform convergence theorem yields t-1 -x6/l (i), and hence if we set v-116 = x(2/pB)1/6 some further manipulation yields the assertion.

We end this section by examining the existence of the r,. Start by observing that

0 -5 qg*(F(s, t)) - qg*(F(0, t)) 5 H(F(s, t)) log [(1 - F(0, t))/(1 - F(s, t))] = O[H(F(s, t))(F(s, t) - F(0, t))/(1 - F(0, t))] = O[t-iH(F(s, t))] = O[1 - (F(s, t))]- 0.

It follows easily that lim,. r(s, t)/Tl(t) exists and hence so does r1 and

r(s) = ris' = s(1 - s)-2 exp (Bgp*(s)).

Assume p <oo. Then (4.3.2) yields r(s) = s(1 - s)-2-B(L((1 - s)-1)-1 and a Tauberian theorem yields

ri ~ il+B"/r(2 + Bz)L(i) (i -- oo). Again ri increases more rapidly with emigration than without. If (4.3.3) holds with 6 <1 then r, will increase faster than any power.

4.4. The supercritical case. In this section we assume a <b so that r> 1. A suitable integrating factor for (4.2.1) is

I(s, 6) = s-1(1 - s)l+~"v(r - s)1-O/v exp (-Bq,*(s))


where v = p(b - a) and q *(s) = f~ H(y)/(r - y) dy. Since l(s, 0) = 0(1/(1 - s)) we see that f(s, O)I(s, 0) --0 as s- 1- when 0 >0 whence

^(s, 6) = (s/pa)(1 - s)-l-/v(r - s)-l+/v[exp (B+p*(s))] (4.4.1) x (1- y) (r y)-/v(ry)- B*e-By) dy.

Throughout this section we assume q*(1-) = oo. This is equivalent to

E(log +Y)= oo and then P,(T < o) = 1 (Pakes (1988b)). We look first at conditions for hi < oo. Let K = B/(r - 1) = a/(b - a)= K/v and

G(x) = 1 - .(1 - e- ).

Theorem 4.4.1. When E log +Y = oo, hi < oo iff

lexp (-K fG(x)dx)

dz < .

Proof. It is obvious from (4.4.1) that h < oo iff

(1 - y)1 exp (-B(p*(y)) dy < 0.

Let z = -log (1- y) in this criterion and observe that -log (1-s)

9Q*(s) = G(x)/(r - 1 + e-x) dx

-log (1-s)

-(r -

1)--1 GG(x)dx (s-- 1),

since p*(1-) = oo. The assertion follows.

Our assumptions are that G(x)-- 0 as x - oo but f" G(x) dx = oo. The criterion of Theorem 4.4.1 may, or may not, be satisfied and the boundary separating the two behaviours occurs when xG(x)-- y 0. When y >0 the criterion is equivalent to f z-Kydz < oo. Thus hi = oo if Ky 5 1 and hi < oo if Ky > 1 and this continues to be the case if y = oo. We shall not pursue in detail the question of the asymptotic behaviour of hi when it is finite because there is no immediate relation with the most general limit theorems for T (Theorem 4.4.2 below). However it is possible to obtain a feeling for the growth rate of hi by considering a couple of special cases.

First let G(x) have the above behaviour and Ky > 1. It is not hard to see that

h(s) - [pa(r - 1)(Ky - 1)]-X[(log (1/(1 - s)))/(1 - s)] whence

hi ~ log i/A(Ky - 1).

If G(x) - cx-^ where 0 < c < oo and 0 < A <1 then h(s) - [log (1/(1 - s))]A/pa(1 -

s) whence h, ~ (log i)A/ppa.


We begin our approach to a limit theorem for T by observing that the transform (4.4.1) can be inverted, as in previous subsections, to give

s (r - 1) exp (vt) (4.4.2) r(s, t) - exp [-B(qg*(F(s, t)) - qg*(s))]

1 - s (r - s) exp (vt) - (1 - s)

where (r - s) exp (vt) - r(1 - s)

F(s, t)= (r - s) exp (vt) - (1 - s)

is the p.g.f. of the size at time t of a subcritical linear birth-death process having birth and death parameters pa and pb, respectively.

Let log x

A(x) = exp G(y) dy

which is differentiable, increasing and A(oo)= oo. The last property is equivalent to our assumption that E(log Y) = oo

Theorem 4.4.2. When b > a and E(log Y) = oo then for 0 < u < 1,

lim P,(A(i)/A(ie•T) u) = uK i--*oo

Proof. As in previous proofs we use P(s, t) = (1 - s)r(s, t). If s = s,-- 1 as t---> oo it is clear that

exp B(q9*(F(st, t) - qp*(st)) - [A(1/(1 - F(s,, t)))/A(1/(l - st))]K.

Observe that xA'(x)/A(x)-* 0, which implies that A is SV at infinity. In addition

r vt+logx

A(x)/A(xe') = exp - G(y) dy log x

increases with x on (1,0oo) from m,, say, to unity and mt,-*0 as t--->oo. Also

A(x)/A(xe') 10 as t oo. It follows that for any 0< u < 1 and all t large enough we can uniquely define y(t, u)> 0 by

(4.4.3) A(y(t, u)) = uA(y(t, u)e").

Clearly y(t, u) ' oo as t 1 oo. Let s, = exp (-6/y(t, u)). Now

1- F(s,, t)-~(y(t, u)e')-1

whence, upon using the slow variation of A and (4.4.3),

It follows from (4.4.2) that P(s,, t)--

uK. The limit is the LST of a degenerate distribution attributing mass uK to the origin and no mass in (0, c). The extended


continuity theorem yields

lim Pxy(t,u), ( T > t)-_> uK (x > 0). t---oo

Let x = 1. The probability on the left can be written as PY(t,u)l(y(T, u) >y(t, u)) and if we let i = [y(t, u)] we have

lim P,(y(T, u) > i) = uK. i--->00

But since A(.)/A(.e'T)

is increasing the left-hand side is P {A(y(T, u))/ A(y(T, u)e'T)) > A(i)IA(ievT)} and the assertion follows from (4.4.3).

Theorem 4.4.2 can be put into more explicit forms by imposing additional constraints on G(x).

Corollary 4.4.1. Let xG (x) -- 0 as K-- oo. Then

lim P,(A(i)/A(e •) _

u) = UK i--*oo

Proof. As i--->oo, T- oo and

f log i+ vT G(y) dy = o(log (1 + vT/log i))

ogi

whence

A(ievT)/A(i) = [1 + vT/log i]•i where Ei---> 0. It follows from the theorem that vT/log if oo and hence

flog i+vT

•i G(y) dy = o(log (1 + log i/vT)))P0. vT

We conclude that A(evT)/A(i)- A(ievT)/A(i) and the assertion follows.

Corollary 4.4.2. Let xG(x)-* y E (0, oo). Then for x> 0,

lim Pi(v T - x log i) = 1 - (1 + x)- K. i---).oo

Proof. Arguing as above we obtain

A(ie•T)/A(i) - exp [y log (1 + vT/log i)] = (1 + vT/log i)Y

and the assertion follows from Theorem 4.4.2 by setting x = u-1/' - 1.

Corollary 4.4.3. Let G(x) be regularly varying at infinity and xG(x)- oo. Then for x > 0,

lim Pi(vTG(log i) x) = 1 - exp (-Kx). i-"00


Proof. A change of variables yields

A(ieT))/A(i) = exp log i G(z log i) dz

_ exp [M(log i) log (1 + vT/log i)]

where M is an arbitrary (large) constant and the inequality holds for all large enough i. It follows that vT/log i-- 0 and hence, using the uniform convergence theorem for regularly varying functions,

A(ievT)/A(i) - exp (vTG(log i)),

and the assertion follows. If yK > 1 in Corollary 4.4.2 or if G(x) = cx-A, with 0 < A < 1, in Corollary 4.4.3,

we see that T/hi has a limiting distribution. If 0 s < 1 then 1 - F(s, t) - const. (1 - F(O, t)), where the constant depends on

s. Using the slow variation of A( -) we obtain

rs r(s,

t)/r(t)-*-->( -- exp (B9p*(s)). (1 -s)(r - s)

Thus the ri exist and as s -- 1

ris'C r [A(1/(1 - s))]K (r - 1)(1 - s)

whence

r,- (A(i))K (i-->

). r-1

We conclude that the stochastic growth of T with i is very slow, i.e., the supercritically growing population is strongly balanced by the very high rate of emigration.

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