non-explosivity of limits of conditioned birth and … of limits of conditioned birth and death...

Non-Explosivity of Limits of Conditioned Birth and Death ProcessesAuthor(s): G. O. Roberts, S. D. Jacka, P. K. PollettSource: Journal of Applied Probability, Vol. 34, No. 1 (Mar., 1997), pp. 35-45Published by: Applied Probability TrustStable URL: http://www.jstor.org/stable/3215172Accessed: 04/06/2010 02:17

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J. Appl. Prob. 34, 35-45 (1997) Printed in Israel

? Applied Probability Trust 1997

NON-EXPLOSIVITY OF LIMITS OF CONDITIONED BIRTH AND DEATH PROCESSES

G. O. ROBERTS,* University of Cambridge S. D. JACKA,** University of Warwick

P. K. POLLETT,*** University of Queensland

Abstract

Let X be a birth and death process on Z+ with absorption at zero and suppose that X is suitably recurrent, irreducible and non-explosive. In a recent paper, Roberts and Jacka (1994) showed that as T - o the process conditioned to non-absortion until time T converges weakly to a time-homogeneous Markov limit, X", which is itself a birth and death process. However the question of the possibility of explosiveness of X" remained open. The major result of this paper establishes that X" is always non-explosive.

INVARIANT MEASURES; QUASI-STATIONARY DISTRIBUTIONS

AMS 1991 SUBJECT CLASSIFICATION: PRIMARY 60J27

SECONDARY 60JG80; 60B10

1. Introduction

We consider the non-explosivity of birth and death processes conditioned not to become extinct. As is often the case with problems involving weak convergence of conditioned processes, it is convenient to consider separate cases depending on whether or not the unconditioned process exhibits quasi-stationary behaviour.

Although it is easy to write down expressions for the limiting process's transition

probabilities in terms of an implicitly defined eigenvalue, it is difficult to obtain more explicit expressions. Therefore Section 5 is devoted to characterizations of the absorption decay rate and its corresponding eigenvector, which (together with the transition

semigroup of the unconditioned process) describe the dynamics of the conditioned

processes. Many of the techniques used here exploit stochastic monotonicity properties which

are unique to birth and death processes. However, some of the results we show can be generalized at least to countable state space Markov processes with each state having only finitely many neighbours.

The theory of quasi-stationarity for birth and death processes given in van Doorn (1991) is essential to our approach.

Received 20 July 1994; revision received 1 November 1995. * Postal address: Statistical Laboratory, 16 Mill Lane, Cambridge CB2 ISB, UK. ** Postal address: Department of Statistics, University of Warwick, Coventry CV4 7AL, UK. *** Postal address: Department of Mathematics, University of Queensland, Queensland 4072, Australia.

35

36 G. O. ROBERTS, S. D. JACKA AND P. K. POLLETT

2. Preliminaries and main results

Let X be a time-homogeneous, non-explosive birth and death process on 7Z+, with up rates 2i, down rates pi, all assumed to be non-zero for i E N. Suppose that 0 is absorbing, so that A = o= 0. Let {P~j(t), i, j E Z+, t > 0} denote the transition probabilities, Pij(t)= P [X,=j I Xo= i].

Define z= inf{t ? 0; X,= 0} and let x,(t)= P[z > t]. It was demonstrated in Roberts and Jacka (1994) that

(2.1) P(t) Xj(T-t) (2.1) pT(t) P[X,= jI Xo =i, > T] =Pj (t) x , t(< T, i, jE -, xi (T)

and that a limiting time-homogeneous sub-semigroup P" given by

(2.2) P, = lim P7 (t)

exists Vi, jE , t O0. However the proof allowed the possibility that P" might be

defective, corresponding to explosivity of the limit. Defining X" to be a process con- structed according to the probabilities P", we shall show (Theorem 3) that, in fact, X" is always non-explosive.

If P [z = o] > 0 then non-explosivity of the unconditioned process clearly implies non-

explosivity for the conditioned process. Therefore we shall assume throughout that < co a.s. This recurrence condition is most naturally stated in terms of the scale function, S, defined by

j-1

Sj = 1 + 1 (ARi7,)-', j > 2, i=S1

Sl = 1,

where rcI= 1, ri = H~-i' Akk1k+1, i > 1. Then S. lim_.

Sj= co if and only if < co a.s. Note that So = co implies that the condition for non-explosivity of X holds:

n=1 j=1

We need to be precise about what we mean by quasi-stationarity.

Definition. If for some i E N,

Pi.(t) t -.' xi(t)

which is a non-defective probability measure, then we say that 6b exists, and is the

limiting quasi-stationary distribution.

Remarks. 1. There are various equivalent definitions. For instance, van Doorn (1991), p. 689,

gives a definition of a quasi-stationary distribution which is equivalent to the existence of a left eigenmeasure of the transition semigroup.

2. If 6~, exists then lim,., P,.(t)/x;(t) exists and is the same for all i.

Non-explosivity of limits of conditioned birth and death processes 37

van Doorn (1991) provides a useful sufficient condition for the existence of 6, together with a link between quasi-stationarity and geometric ergodicity. Define

V= I nTn

' I 7rii n=1 i=n+l

and introduce the abscissa of convergence of P,

{=sup J: e sPj(s)ds < oc .

The solidarity property of a (independence of i,j) is well known; see for example Jacka and Roberts (1995), whilst it is clear that a > 0.

Theorem 1. (van Doorn 1991). (i) V < co o> a > 0 and (ii) 6, exists a > 0.

We will require a minor extension of this result.

Theorem 2. The following are equivalent. (i) For some i, j E N

P, (t) _.. ~-( some non-zero limit.

xi(t)

(ii) 6 exists.

(iii) a > 0.

If none of these hold then P, j(t)/x, (t) -+0 Vi, jE as t - o0, whilst if they do then a =

Remark. Since the neighbourhood of 0 (the set of states which can exit directly to zero) is finite, it follows that a= p where y is the abscissa of convergence of x,(t):

P = sup {; e'sxi (s)ds < }o .

This was shown in Jacka and Roberts (1995). Therefore (iii) is equivalent to P > 0.

The main results of the paper are as follows.

Theorem 3. X" is a non-explosive birth and death process.

Corollary 4. If 6, does not exist (which is equivalent to a =P = 0), then the limiting process is a birth and death process with rates given by

Si' S

3. Links with quasi-stationarity and monotonicity

Since the non-existence of 6, will play a major role in our approach, we begin with a preliminary result which uses stochastic monotonicity properties of birth and death processes.


Recall (Roberts 1991) that a sub-semigroup P is strongly stochastically monotone if, for all possible choices of il < i2, P,2 (t)/Pi,j(t) is a non-decreasing function of j Vt > 0. It is well known that birth and death processes are (essentially) the only class of Markov

processes on Z+ to be strongly stochastically monotone (Karlin and McGregor 1959).

Lemma 1. Pll(t)/xl(t) is a non-increasing function of t.

Proof. For tl < t2, we can write

PlJ(t2) = I Pli(t2 - tl)Pij(tl)

i>1

Plj(tl) = I Ii(i)Pj(tl) i21

where I,(j) is the indicator function of 1. Recall the FKG inequality (due to Fortuin

Kasteleyn and Gimbre, and used in the context of conditioning in Roberts (1991)): if

fi, f2, gl, g2 : ~ rN are non-negative functions such that f21fl and g21lg are non-decreasing (but possibly infinite), then

2(i) f2)2 (i)gl(i) Pfl

2(i) f2(igl(i).

Now, setting fl(i) = I,(i), f2(i)= Pli(t2 - t1), gl(i)= Pij, (t), g2(i)= Pij2 (t) (with j, j2), the conditions for the inequality follow easily from strong stochastic monotonicity, so that,

applying the FKG inequality above, we obtain:

P (tl)Plj2(t2) f(i)g2(i) f(i)g(i)

> f(i)g2(i) f2(i)1(i) = Plj2 (tl)Pl, (t2),

for j, j2. If we set j, = 1 and sum over j2 ? 1 we obtain Pll(tl)XI(t2) ? Pl1(t2)x1(t1), which

rearranges to give the desired inequality.

Proof of Theorem 2. The equivalence of (ii) and (iii) is Theorem 1, and clearly (ii) implies (i). We shall therefore complete the proof by showing that (i) implies (iii).

Let Q be the Z+ x Z+ Q-matrix for X. Then, since Q is non-explosive, P satisfies the forward equation

dPtJ

=d (PrQ)j0, i, jE if.

In particular for j = 0 and i e N, it follows that

(3.1) x(t)= ~ [z > t]= 1- o(t)= exp - G•(1)plds ,

where 6 (1)=P,(l(s)/Ix(s). Now if condition (i) of Theorem 2 holds, by solidarity and Lemma 1,


Pll(s) lim > 0, s--.

o+ XI(S)

and from Jacka and Roberts (1995),

- log xI(t)

-= #= lim S-mO t

so that, from (3.1),

Pll(s) (3.2) a 0 =, lim > 0 s -. c XI(S)

as required. The final results in Theorem 2 now follow from the failure of condition (i), and from equation (3.2) respectively.

4. The conditioned processes

We will make use of the condition for explosiveness of a birth and death process (derived by construction from Reuter's more general explosivity condition): an irreducible birth and death process on 7Z+, with birth rates a, and death rates bi, is non-explosive if and only if

i= 1 P j=o where P,= i• o ailbi+l1, i E N, Po = 1.

To avoid tiresome details of comparing sums with different starting points, we shall state the following trivial lemma immediately. We will use it repeatedly without further comment.

Lemma 2. If {c,}, {d,} are sequences of positive numbers, then Z cn cZn

I cn I d, = co : I Cn did= co n=1 = i=0 n=nl i=no

for n1 _2 1, no ? 0, no <=

rtn.

Recall (for example from Jacka and Roberts (1995)) that a vector / is a-subinvariant for P if

(4.1) (P(t)f3). <e e-'"fl , iEN Vt 0, and it is non-negative. If equality holds in (4.1), then # is said to be a-invariant (otherwise, strictly L-subinvariant).

Theorem 5. (Roberts and Jacka 1994). Let XT be the conditioned process, [X I z> T]. (i) XT has time-dependent rates given by

2T(t) = 2; xi+,(T- t)/x, (T- t), pfr(t)= pixil(T-

t)/x; (T- t).

(ii) The 2T(t) (respectively pr (t) are non-decreasing (respectively non-increasing) asfunc- tions of T, and converge to finite non-zero time-homogeneous limits, ~i" (respectively p7).


(iii) Denoting lim,•,

x,(t)lxl(t) by ,3, / is an a-subinvariant vector for P.

(iv) If p is a-invariant, then {(X'), T =

0} converges weakly to (X'), and X" is non-

explosive.

It is clear from (i) that 2• = r/A + i, 1/fi, = pi, /, - 1/f,, and so, by a direct substitution of Reuter's condition, X" is non-explosive if and only if

(4.2) 01 j =7 . i=1 i7 ij=1 i i+

We shall prove that if V= co then (4.2) holds. In what follows, denote by Q the restriction of Q to N.

Lemma 3. P is an a-eigenvector for Q, i.e. Q = - a. Moreover, / is the unique a- eigenvector for Q with /p = 1.

Proof. x(t) satisfies the backward equation

dx(t) Ox(t)- dt

i.e.

dx(t) (4.3) A, x, _ (t) + A, xi ,+(t) -

(, + A,)x, (t) - , i 1. dt

Therefore from (3.1) and (4.3) we obtain,

xi-_(t) xi+,(t) x,(t) x,(t) (4.4) (p, + ( + p,) =- 6M(1)p,- x(t) xW(t) x,(t) x,(t) and from Theorem 2, limt,,, 6'(1)1 =a. Taking limits in (4.4) gives

(4.5) p; f;-1 + i A +I - (A, + P ) # = - afp,.

Thus, since x0(t) =0 and, hence, lo= 0, / is an a-eigenvector for Q. Finally, it is clear that (4.5) has a positive solution which is uniquely determined by PI (see the proof of Theorem 6).

Remark. We have shown that P is an a-eigenvector of Q. Of course general theory (see, for example, Theorem 2 of Pollett (1986)) does not guarantee that P is a-invariant for P, which would then permit the application of Theorem 5. However, we can immedi-

ately prove the following result.

Lemma 4. If a > 0 then # is an a-invariant vector for P.

Proof. First observe that Q is symmetrically reversible with respect to it, i.e.

Thus, since / is an a-eigenvector for Q, the measure m, given by m,

= i, /3, i e J, is a (left) a-eigenvector for

Q, i.e. mQ= -a m. It follows, from Theorem 3.2 of van Doorn


(1991), that i mi = ~xle; notice that, since # is an ca-eigenvector for Q, we can identify /, with Q, (c), where { Q, } is the 'Karlin and McGregor' sequence of orthogonal polynomials defined by van Doorn's system of equations (2.9). Since m is summable, Theorem 3 of Pollett and Vere-Jones (1992) implies that m is also ca-invariant for P, i.e.

(mP(t))j = e-"'mj, jE N, t > O.

But, by Theorem VII of Kendall (1959), P is also symmetrically reversible with respect to 7r, i.e.

ri Pij (t) = •j

P;i(t), i, jE N, t =0O,

from which it follows immediately that P is an a-invariant vector for P.

We shall extend the result of Lemma 4 to the case when = 0 by using the following estimate for p.

Lemma 5. Let yj be the solution to QTy = 0 with the boundary condition y, = 1. Then,

V1 _j ? i,

/ <LT /3j -= T

Remark. We chose the boundary condition to be consistent with the definition of fl.

Proof. From (4.3), Qx _

0 componentwise since x,(-) is clearly decreasing. Define

yi (t) = xi (t)lxi _ l(t).

Then

(4.6) Aiyi+l(t)+- =

i yi(t)

Now, we shall prove by induction that

(4.7) 0 yi(t) O 8_, Vi > 2,

where Oi = y/ly- 1i.

Suppose 0 ? yi(t) ? 08. Then - 1/y(t) ? - 1/0, and from (4.6), 1iyj+l(t) ?

Ai + pi [1 - (1/Oi)]= iOi+ ,,. So

(4.8) 0 ? y (t) ? O8, i 0 y+,I(t) . Oi+l.

A similar but simpler argument establishes that 0 ? y2(i) - 02, so by induction, (4.7) is

proved. Now,

lim y, (t) = , t-- ociA-1

so that in the limit

-i-1 V-- i

and the result follows.

Remark. In particular, it should be noted that Pi _

y,. Moreover, y (defined in Lemma 5) can be identified with the scale function S.


Lemma 6. y = S.

Proof. Letting zj = j- y,, for j > 2, we have - pjzj + 2j zj,+ =0, which implies that

Zi = z2 fl fl pi

i=2 - i i= i i

so that

i j-1

yj = y, + zi = 1 + 1 i=2 i=1 i i

Lemma 7. V= co -> X is non-explosive.

Proof. Lemmas 5 and 6 can be used to demonstrate that

i= "1=i ij=1 fifi+l i= 1 i j=1 Si S,+

Denoting the right-hand side by E, Reuter's condition implies that E= co is sufficient for non-explosivity of X". However we can write

E I 1j S[i - I+

j=1 i=j S Si i=

1 j=1 S i S• +I,,

and the interior sum equals S•'. Therefore E= p-1' 1 7jSj. However, by interchanging the order of summation for V we can show that

E= V+ pyl' Z Icj. j=1

Thus V= co implies that E= co, and the result follows.

Proof of Theorem 3. If = 0 then V= oo and so, by Lemma 7, X" is non-explosive. Now suppose that a > 0 (equivalently, 6, exists). Then, by Lemma 4.1 of Jacka and Roberts (1995), we have

xl(T- t) T- e t > 0. xl(T)

It follows, from (2.1) and (2.2), that the semigroup, P", of X" can be evaluated as

S(t) = e"'Pj (t), i, jE t 0.

But, by Lemma 4,

SPii(t)njo=e-'-i'd, iE e.

Hence P" is non-defective, i.e. X" is non-explosive.


Proof of Corollary 4. Since c = 0 we have that / = y. The result now follows immedi-

ately from Theorem 3, Theorem 5(ii) and (iii), and Lemma 6.

Remarks. 1. When 6. exists (equivalently, a > 0) it is given by

6•(i)= ap•-I'tifli, for i E N; van

Doorn (1991) gives related but not identical results. 2. It follows that, for every a > 0, a and # bear the following relationship:

iEN

with the interpretation that c= 0 when and only when the sum diverges.

5. Characterization of a and # for birth and death processes

Since all the formulae given so far only give expressions for ( and # in terms of one another, the following characterizations may be useful. Once again they rely heavily on

monotonicity properties. The first result identifies a probabilistic representation of r- invariant vectors for 0 ? r < a.

Theorem 6. (Characterization of fl). Define

1 (5.1) fir(i) =

l(5.1)[eItI[ < oo]]

where z; = inf{t > 0; X, = i}. Then flP is the unique solution of Qy, = - ry, with y (1) = 1.

Proof. It is clear from the results of Jacka and Roberts (1995), that l, (i) > 0 Vi E -. Uniqueness of the solution follows from the fact that we may regard a solution y, as

satisfying the second-order (non-autonomous) linear difference equation Qy, = - ry, with the initial conditions y, (0)= 0 and y, (1)= 1. So all that remains is to show that f3, satisfies the equation. Now

1 =

[El[erri+ I[Ti+, < 00]] /3r(i+ 1)

1 -= i)[e"'+ I[Tz+, < oo]]

fr(i)

= I3r00[e--r)t [A + j E-I-[er?+' I[z, 1 < oo]]]]dt. fr(i)

So, /r(i) -r + Ii (i- 1)

/r(i+1) q-r q;-r r(i+ 1)'

and rearranging the terms gives the required expression. In particular, for r= a we know from Lemma 3 that /3 is the unique solution of

Q/ = - a/ with /3(1)= 1. Therefore /3 = ,.


Theorem 7. (Characterization of a). The following are equivalent expressions for a. (i) sup{r : 3 a positive solution to Q = -r/ with /(1) = 1}, (ii) inf{r : a positive solution to Q/= -r/ with i(1)= 1}, (iii) sup{r : 3 a non-decreasing solution to Qi =- r/ with /(1) = 1 }, (iv) inf{r: A a non-decreasing solution to Qi = - r with (1)= 1 }.

Proof. van Doorn (1991) establishes the first two characterizations; we prove these for the sake of completeness.

Let us denote the expressions in (i)-(iv) by a,?I, O4, respectively and let Yr be the

unique solution to

(5.2) Qr= -rYr with ,(1)= 1.

Now it follows from (5.2) that the process er•'f#(X,) is a local martingale. By Theorem 6, ,r is positive for r a. However, suppose that for some r > a, y, is positive. Then

er'Pf, (X,) is a supermartingale (since a non-negative local martingale is always a supermartingale), so that y, is r-subinvariant for P. However no r-subinvariant vectors for P exist with r > a (Seneta 1981), and therefore no non-decreasing r-subinvariant vectors for P can exist with y, (1)= 1. It follows that tl = = , and that 22, 94 < since / = # , is non- decreasing and positive (as we shall see below).

Since y, = flp > 0 is the unique r-subinvariant vector for P, for each 0 ? r a, X2 = L. It remains to prove that for 0 ? r a, flp is non-decreasing. However letting Ai= r (i)- flr(i- 1), we obtain (from (5.2)) Ai Al - y Ai = -rpf(i) 0. So that, letting G.=

(5.3) G+l

< G,,

since 7ri = 1 Ai -_l 1. It follows that G is non-increasing and we can write

/,r(i) = 1+ Aj j=2

Gj = 1 +- Cl j=2 '~-j-i-I1

Then, if fr is not increasing, 3i E N and a c > 0 such that Gi < - c. Therefore we can write,

i-1 k G. fr(k) 1 +kL I Aj +L

lIZ j=2 j=i j-_1

-j

k-I 1

<flr(i)--CL

Z

However X7=i

l/,j rj = oo, so that such a •, will eventually become negative for sufficiently

large arguments. This contradicts the characterization in Theorem 6, and therefore •, (.) is non-decreasing for 0 ? r ? a. Thus

o4 = a also.


References

JACKA, S. D. AND ROBERTS, G. 0. (1995) Weak convergence of conditioned processes on a countable state space. J Appl. Prob. 32, 902-916.

KARLIN, S. AND MCGREGOR J. (1959) A characterization of birth and death processes. Proc. Nat. Acad. Sci. 9, 1141-1164.

KENDALL, D. G. (1959) Unitary dilations of one-parameter semigroups of Markov transition operators, and the corresponding integral representations for Markov processes with a countable infinity of states. Proc. London Math. Soc. 9, 417-431.

POLLETT, P. K. (1986) On the equivalence of p-invariant measures for the minimal process and its

q-matrix. Stoch. Proc. Appl. 22, 203-221. POLLETT, P. K. AND VERE-JONES, D. (1992) A note on evanescent processes. Austral. J Statist. 34, 531-536.

ROBERTS, G. 0. (1991) A comparison theorem for conditioned Markov processes. J Appl. Prob. 28, 74-83.

ROBERTS, G. O. AND JACKA, S. D. (1994) Weak convergence of conditioned birth and death processes. J Appl. Prob. 31, 90-100.

SENETA, E. (1981) Non-Negative Matrices and Markov Chains. Springer, Berlin. VAN DOORN, E. A. (1991) Quasi-stationary distributions and convergence to quasi-stationarity of birth-

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