On a Functional Central Limit Theorem for Markov Population...

On a Functional Central Limit Theorem for Markov Population ProcessesAuthor(s): Andrew D. BarbourSource: Advances in Applied Probability, Vol. 6, No. 1 (Mar., 1974), pp. 21-39Published by: Applied Probability TrustStable URL: http://www.jstor.org/stable/1426205Accessed: 04/06/2010 19:40

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Adv. Appl. Prob. 6, 21-39 (1974) Printed in Israel

? Applied Probability Trust 1974

ON A FUNCTIONAL CENTRAL LIMIT THEOREM FOR MARKOV POPULATION PROCESSES

ANDREW D. BARBOUR, University of Cambridge

Abstract Let {XN(t)} be a sequence of continuous time Markov population processes

on an n-dimensional integer lattice, such that XN has initial state Nx(O) and has a finite number of possible transitions J from any state X: let the transition X-- X + J have rate Ng'(N - IX), and let gJ(x) and x (0) be fixed as N varies. The rate of convergence of ,/N (N- IXN(t) - (t)) to a Gaussian diffusion is investigated, where $(t) is the deterministic approximation to N-IXN (t), and a method of deriving higher order asymptotic expansions for its distribution is justified. The methods are applied to two birth and death processes, and to the closed stochastic epidemic. MARKOV POPULATION PROCESS; FUNCTIONAL CENTRAL LIMIT THEOREM; EPIDEMIC; COMPETITION PROCESS; BIRTH AND DEATH PROCESS

1. Introduction

A common method of describing biological and ecological systems is to represent them as continuous parameter Markov processes, in one or more dimensions, with each coordinate typically representing a count of the present population of a

particular species. The transition rates in these models are usually simple functions of the coordinates, and sometimes of the time parameter as well. We shall consider a sequence

{XN(" )} of such processes, in which the indexing parameter N deter-

mines the magnitude of the initial state of the process, in that XN(O) = Nx(O) for a fixed vector x(O). In Barbour (1972) [B] a heuristic argument is put forward, which leads to a method of successive approximation, in terms of large N, to the

process xN(t) = N-'XN (TN(t)), where T,(t) is a suitably chosen function of t: see

also McNeil and Schach (1973). The approximation, which to the second order

reduces to a small Gaussian diffusion about the deterministic path, was found

effective in the example chosen even for relatively small values of N. The limiting

process has much to recommend it as a natural choice for approximation, and

even the apparently restrictive initial condition requirements proved amenable in

practice. In this paper, we provide a rigorous justification for these ideas. The basis for

the argument is to be found in Kurtz (1970) and (1971), where the first and second

Received in revised form 26 June 1973.

21

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22 ANDREW D. BARBOUR

order approximations are justified under very weak conditions. In order to take full advantage of the method of [B], we will consider only a more restricted situation, which nonetheless includes all of the most common biological models, although much of the material can be proved in a more general setting.

Section 2 consists of a statement of Kurtz's functional central limit theorem, in a form compatible with the aims of this paper, and in Section 3 an associated rate of convergence result is proved. Section 4 contains a detailed study of the properties of integrals of the deterministic equations. The application of the method is illustrated in Section 5 with some examples.

2. Basic elements

For some fixed T> 0, let xN('),

N = 1,2, ...,

be a sequence of continuous parameter Markov processes on [0, T], taking values in an n-dimensional lattice

LV = N-'Z"; let their stable, conservative Q-matrices be specified by the transition intensities

(2.1) x-+ x + N-'J at rate Ng'(x)

for J in some fixed finite set f c Z"\ {O}, and let their common initial value be x(O), independent of N: Z here represents the integers. Assume that the deterministic solution to the process exists in [0, T], that is:

Assumption A. There exists a unique function 4(t), 0 < t < T, satisfying

(2.2) 4(t) = E Jg'(4(t)), 0 < t t

T; 4(0) = x(0).

Let S c R" be defined by

(2.3) S = {x: x = (t) for some 0 t< T},

and, for some fixed e > 0, define

(2.4) Se = {x: IX - S 8e}. Assumption B. The functions g'(x), J ef, are multinomial in the coordinate

variables x1, x2, ..., xn within S

. Note that, when verifying Assumption B in applications, e may be taken as small as need be.

Let constants L and M,, r = 2, 3, ..., be defined by

M, = sup 1 gJ(x)jJr', r = 2,3, ..., xeS8 J

L = s up[ J(g(x)- g(y)) / x .

x,y e S L 1 Let yN(t) - N ){x )}, and let gf(x) denote

@g'(x)/@xj.

A functional central limit theorem for Markov population processes 23

Theorem K. Under Assumptions A and B:

(i) for any 0 < 6 e, 0 < t < T,

(2.5) P [sup IxN(s)- (s)4 > J

I N16-2tM2exp(2Lt);

(ii) given any integer r _

3, there exists a constant Cr depending on e, T, L and Mk, k = 2, 3, ..., r, such that

(2.6) P sup !XN(S) - 4(s) > ?8 CrN'-r

LO:_s<5T i

(iii) YN => y in D[0O, T], where y(t) is the diffusion with y(O) = 0, whose charac-

teristic function F(O, t) satisfies

(2.7) -t I (0.J) g(1(t)) + 21 (.J)2g'(4(t))'D = 0. J j=1 jj 2,

Remark. This is a special case of Theorem (3.5) of Kurtz (1971) [K]. The developments that follow can be modified without difficulty, but at the expense of greater detail, to more general situations. For example, similar results hold if g'(x) is replaced by g (x, t), where gY(x, t) is continuous for each J as a function of t, uniformly in x r S' and in N, and where g '(x,t) converges uniformly in S' x [0, T] to g-(x, t) as N -+ oo. Assumption B is only made to guarantee once and for all the existence of as many derivatives of the functions g' as are necessary, and it can be weakened in the obvious way in Theorem 1. It is also possible to improve upon (ii) with an estimate of the form

P[ sup xN(s) - 4(s)I > e] ? Cexp(- cN4).

3. Rate of convergence

Let h(x, t): S' x [0, T] -* R be any solution of the equation

Oh Oh (3.1) +

.

g'(x) h

Jk = 0 t k @Xk

which has all its second partial derivatives with respect to the coordinates x , and also the mixed derivatives @2h /xxjt. Any such h is an integral of the deterministic equations, and is constant along deterministic trajectories. Let Vh = Vh(4(t), t) denote the vector of derivatives of h(x, t) with respect to x, evaluated at x = 4(t).

Lemma 1. The random function z: z(t) = y(t). Vh(4(t), t), 0 < t < T, where

y is the diffusion limit specified by (2.7), is distributed in D[0, T] as a Wiener process deformed by a non-random change of time scale.

Proof. From Equations (2.2) and (3.1),


d (ah =

a2h d4k a2h -1 + -- dt j - k jk dt jt

(3.2) = (x) I Jk O ag kh

- x , J k k x= ?

t I j k k

- Jk

J 0? k k

Let tP,(), 0 & to < t < T, denote E{exp(ifz(t))}, where E is the expectation conditional on an arbitrary given distribution for y(to), whose characteristic function is denoted by Pto(f).

Then

(3.3) d -= i-E exp(iVz(t))y(t). dVh }+ dtE{exp(iO. y(t))}]0Vh

and substituting in (3.3) from (3.2) for dVh /dt and from (2.7) for the second term, we obtain, after integration,

(3.4) Yt = Ptoexp{

- @12 t2(u) du ,

where

(3.5) a2(t)= g((t))(J.Vh)2.

This identifies z through its finite-dimensional distributions.

Now define zN(t)= YN(t).Vh(4(t),t), and let PN(, ,t) and Y(0, t) be the characteristic functions of zN(t), z(t) respectively, conditional on yN(O) = y(O) = 0. Let E2(t) = fta2(u) du, where a2(t) is defined by (3.5), and set

(3.6) H = sup IVh((t),t)I; Ot<T

note that, from (3.4), z(t) /l(t) is distributed as a standard Normal random variable.

Theorem 1. In addition to Assumptions A and B, and assumptions about the smoothness of h, suppose that

(3.7) inf a2(t) = 2 > 0.

Then, for any fixed t, 0 < t < T,

(3.8) sup I P[ZN(t) < v] - (v /l(t)) = O(N -?log?N),

where the order term depends on h, t and the parameters of the process, and where

Q(.) is the distribution function of a standard Normal random variable.


Proof. The error estimate is based on an estimate of the discrepancy between the characteristic functions N, and '. For convenience, we work with the process XV, which is obtained from the original process XN by stopping it when it first leaves S": then xN(t) E Se almost surely for all N > 2 max J I e, and yet, because of (2.6), the estimate (3.8) is unaltered by this modification of the underlying process outside S3*.

Le I(x) be the function which takes the value 1 on S I and zero otherwise, and take N > 2 max Jj/e. Let kj, j = 1,2 **-, denote constants independent of N and t, but depending on s, T, H and the parameters of the process. Let E denote expectation conditional on yN(O) = 0, set Q(u) = e"- 1 - u - u2, and let

g9 denote ag'(x) /xk evaluated at x = 4(t);

x denote 4(t) + N-1-N(t), or equivalently x,(t);

4 denote 4(t);

and replace XN by xN in earlier definitions. Define

pN(x, t)= P[x,(t) = x lXN(O) = x(0)]; then the Kolmogorov forward

differential equation for the process x, takes the form

pN(x, t)/ lt = N ,

(pN(x - JN -', t)g(x - JN-')I(x - JN-')}

- NpN(x, t) g'(x)I(x),

for any state x accessible to xN. There are only finitely many such states, since the process is confined to S' n L", and so the equations have a unique solution; furthermore, multiplying them by exp(i0. x) on both sides, for any fixed 0e R", and summing over all accessible states x,

d dE {exp (iO. XN(t))}

(3.9) = N (exp(iO. JN-') - 1)E{g'(xN(t))I(xN(t)) exp (iO. N(t))}

since all the expectations exist. For each real f,

d d

[d E{exp(M. Y d iE texp(iizN(t)) YN(t). dt•

+ dt E{exp(. (t))}] = Vh

Substitute from Equation (3.2) for dVh /dt, and expand the second term using Equation (3.9), expressed in terms of yN(t) = N*(xN(t)- 5(t)). This leads, after some manipulation, to


dYWN dt + ?+2(t)2 T2N

= N T (exp(iiJ. VhN-) - 1)E{exp(ilzN(t))g'(i) [I(xi

- 1]} J

- 1?2 (Vh .J)2E{exp(ilzN(t)) [g'(9 ) - g,()]}

(3.10) + i/

, J. VhN4E{exp(i0/zN(t)) [g'(i) - g9(5) - (, k) 9] I k

+ N Q(i/J. VhN-+)E{exp(i0zN(t))gJ(x)}.

Let

(3.11) r(t, t, N) = N ,

Q(iIJ " VhN-)g"(4),

J

and let

(3.12) R(/, t, N) = r(/, u, N) du;

note that

Ir(, t, N)I - N I J. VhN-[3 sup g'(x') (3.13) x'

E•se

_< k,1 13 N-*,

and hence that

(3.14) R( , t, N) I < k, I 13N-+t.

Split the last term in the R.H.S. of (3.10) by setting

g'(x) = g'(4) + (g'(x) - g(4))

and take the part involving g'(4) alone onto the L.H.S. This gives an estimate

d + If2a2N(t)N - r(N/, tN)'NI

(3.15) ? I Ik2NIP[ N(t) 1 Sk] + V' E2k3E lx(t) - 4(t) + I IIk4N4EIiN(t) - _(t)I2 + I k/ I3k5N-*ExIN(t) - 4(t)I.

From Equation (2.5), it follows easily that

EI iN(t) - %

(t)I ? kN-a , (3.16)

El 4(t) - 5(t)12 _

kN-ilogIN.


Thus the R.H.S. of (3.15) is bounded by

(3.17) k8 IN-logeN +:k 2 12N- + kol 13N1 K(/, N).

Now TP(I, t) = exp { - ?f2E2(t) + R(/, t, N)} satisfies

(3.18) d

+ ?t2/2a(t) T - r(i, t, N)Tv* = 0,

and tV(,, 0) = 1 = ~N,(I, 0) = '(0,O). Let

(3.19) M = 12/4k,:

then, if I =

MNM, it follows that

'P (C, t) - T'(/, t) j = exp ( - {,i2E2(t))

exp (R(/, t, N)) - 1j

(3.20) < k, I I IN--texp{- ?tzr22}. Also, if AN,(, t) - (T' (, t) - (\ ', t),

(3.21) AN(C, 0) = 0

and (3.22)dA (3.22)

dt + (&22(t) - r( t, N))AN ? K(t, N).

Integrating (3.22), for li/ ? MNM, gives

AN(C0, t) j K(, N) fexp {2(1i2(u) -

_l2(t)) - R(, u, N) + R(i, t, N)} du

(3.23) < 4(z/)- 2K(i, N)(1 - exp(- ?/2z2t)).

But ~,N(, t) is the characteristic function of a probability distribution with finite mean, and P(/, t) is that of N(0, E2(t)). Hence, applying Chung (1968), p. 208, Lemma 2, we deduce that

sup I P[zN(t) ? v] - -(v/IE(t)) 2

2

4fo

4v (3.24) <

-- ( I A(0, t)j1 + Ij V (I, t)

-- (", t) )-i'd

+ 24N -(M E(t) /(27r3)) .

The integral is estimated using (3.20), (3.23) and (3.17), and the fact that, for all real w, 1 - exp(- w2) ? min(w2, 1): the theorem follows.

Remark. It is easily possible to calculate an explicit formula for the upper bound of the error (3.8), in terms of the parameters of the process. It is omitted here for clarity, and because, in practice, the accuracy of the approximation seems vastly better than the bound suggests.


4. Further properties of the deterministic integrals

One of the main points of the method of [B] is that it shows how to get a more accurate asymptotic description of the process x than that given by the central limit theorem. The essential feature in the argument is a sequence of functions d0,(x,t), r = 0, 1, .-, such that Yf=oN-i~j(x,t) is in some sense a martingale, but for an error of order N-'-'; the appropriate functions to take for 00 are integrals of the deterministic equations, and the functions Oj, j > 1, can in principle be calculated sequentially. The discussion in [B] is heuristic, but it can be made precise on the basis of the following paragraphs.

Let q(x, t): R" x R + -+ R be measurable, and t-differentiable in t ? 0 for each x. Define

(4.1) Q,(x, t) = a/(x, t)/at + N ,

g'(x) {q(x + JN', t) - P(x, t)}.

Let z = inf{t: t 0, O,x(t)4 S+e}, and set i = min(t, T). Let Ft

denote the a-field generated by XN(S), 0

= s < t; let U denote S" x [0, T], and C(U) the set of

functions q(x, t): U -+ R which are infinitely differentiable in x, and once differentiable in t.

Lemma 2. Let

ZN(t) b(xN(ft), t) -f Q4(xN(s), s)ds:

then the family {ZN(t),.,,O t ? T} is a martingale.

Proof. Because the event {z < s} is •,-measurable, it is enough to show that

(4.2) E{ZN(t) I,} = ZN(s), whenever XN(s) eS "n0 L .

This is easily verified, because xN(t) can only take one of a finite set of values, either by semi-group methods, similar to those in [K], or directly by using the forward differential equations.

The sequences of functions {k,} of [B] are constructed as follows. Po(x, t) is taken to be any solution of Equation (3.1), or, equivalently, any integral of the deterministic equations

(4.3) x = X Jg'(x).

The functions 0, are then determined recursively from the formula t r+1

(4.4) r(x, 1) = - L( L,(,r+ 1 -.,) ((w), w) dw,

where

(4.5) L,(k) (x, t) - r

g'(x) Hi Jj; a/xj, p(x, t), J j ***,jr=l m= 1


and where qr(w) = -q(x, t, w) is that deterministic trajectory which has q(t) = x: thus, x(u) = q(x, t, u) satisfies (4.3) together with x(t) = x. Under Assumption B, it is always possible, for some E > 0, to find functions o0(x, t) in C(U), and only such functions will henceforth be considered. A set of n such independent functions (which can be taken as a natural coordinate system for the process) is given by the components of the vector q(x, t) - q(x, t, 0): see Hartman (1964), chapter V. This, and the form of (4.4), implies that the sequence can be continued for any number of terms, and that f,(x, t) =- j=o N-kj(x, t) is in C(U).

Lemma 3. For any 0 ? s < T, let t* be a stopping time such that s ? t* < I almost surely. Then

(4.6) E{,r(xA(t*), t*) I~,} = Ix,(x(s),s) + O(N-'-1),

where the order term is independent of t* and ?,.

Proof. The construction of r was chosen precisely in such a way that

Q,,(x, t) = N-'r 'f(x, t, N), where, for each (x, t)E U, f(x, t, N) is of order 1 as N co. In fact, Expression (4.4) ensures that the equations

r+1

(4.7) 80,(x,t)/8t + Z Ls(,r+,1s)(x,t)

= 0

are satisfied for r ? 1, and Equation (4.7) is also satisfied for r = 0, when it reduces to Equation (3.1). This gives

Qo,(x,t) t =0N-

j(x't)) + N g'(x) N-'{o(x + JN-1, t)- O(x, t)}

(4.8) J i=o

+over U. But the coefficient of N- in (4.8), j = 0, 1, , vanishes becaus(N-r-e of (4.7),), j=O IS=1

where the order term is the sum of Taylor remainders in the estimates of dif- ferences 4j(x + JN-1, t) - Oj(x, t), and hence, since each 4j is in C(U), is uniform over U. But the coefficient of

N- in (4.8), j = 0, 1,

., vanishes because of (4.7),

and so

sup If(x,t,N)f = 0(1) as N-* coo. (x,t)eU

The lemma now follows from Lemma 2 and the optional stopping theorem, since the corresponding Z martingale is bounded in U.


Lemma 4. Let h(x, t) e C(U) be any integral of (4.3), such that h(x(O), 0) = 0. Then, for any r

_0, (4.9) Ej h(xN(t*), t*) I' = O(N-"') independently of t*,

where t* is defined as in Lemma 3.

Proof. It is enough to prove (4.9) for arbitrary even integral r. Let 0 without suffix denote a generic member of C(U), not necessarily the same at each ap- pearance. Then, if / = Ohk, L.(!) = Ohk-s for each integer s, 2

_ s ? k. Hence,

from (4.4), constructing the sequence {0,} starting with •o

= h',

1(x, t) -= L2(h)(q(x, t, w), w)dw

- h- 2(x, t)f kO(q(x, t, w), w) dw,

since h is an integral of (4.3). Proceeding by induction, using (4.4), it follows that, for all 0

_ s

__ r,

(4.10) 0,(x, t) = p(x, t) [h(x, t)]r 2. Hence, applying Lemma 3 to ',(x,t) = , o=oN-J'j(x,t),

and using (4.10), we have

(4.11) E[h(xN(t*), t*)]r'=O

•=

N-jE[h(xN(t*), t*)]r- 2j, even r> 2,

where the order term depends on t* only through the expectations. Equation (4.9) now follows, for all even r > 2, inductively from (4.11).

Remark. From Lemma 3 with r = 0, it follows also that

E h(xN(t*), t*) = O(N- ).

Taking the components of l(x, t) lq(x, t, 0) - x(0) as the functions h in Lemma 4, it is easy to deduce that, for all r > 0,

(4.12) ElTq(xN(t*), t*,0) - x(0) I' = O(Nr-f).

Now, from the definition of L and the fact that x(u) = q(x. t, u) satisfies (4.3), it follows that, for any te [ 0, T] and any x', x2 such that il(x', 0,v)e SE for all 0 ? v V t,

Sq(x', 0, t) - q(X2, 0, t)0

SIx,--x'2 IX2+ j gZ(1(X1,Ov)) - _2gJ(q(x2,0,v))}dv

Sxt - x 2 f LIIx,0,v) -(x2,O0,v)Idv. fo*{


Thus, from Gronwall's inequality,

I(x' 1,0, t) (x2,0, t)I Ix' x2 eLt, and so, taking rq(xJ, t, 0) for xj,

I x - x2 I1,(x', t,O) -,q(x2, t,O)e*, provided that q(x2, t, v) e S' for all 0 < v ? t. It now follows easily from (4.12) that

(4.13) ElxN(t*) - _(t*)f' = O(N-'r),

the order term depending on t* only through T. Lemma 3 and Equation (4.13) are between them enough to justify the detailed

calculations in [B], insofar as the distribution and moments under investigation belong to xN(f); but taking t* = i in Equation (4.13), it follows that, for any r > 0,

(4.14) P[f < t] = P[I x,() - (f1) > Je] = O(N--'),

so that the distribution of xN(i)is an accurate asymptotic approximation to that of xN(t). Thus, although moments evaluated for xN(f) may not be those of XN(t), any distributional information obtained from them applies, within the limits of Equation (4.14), to xN(t) also.

The results of Lemma 4 can be complemented as follows. For h as in Lemma 4, let

H = sup jVh(x,t)'; (x t)eU

W = sup X X JJJk,2h(x, t)/8xj8xk (4.15) (x ) Eu J k

G = sup gJ(x)IJj2; (x,t) EU J

W = sup gCJ(x)W7; (x-r)E U J

c = Hmaxl J. Lemma 5. For any k > 0, under the assumptions of Lemma 4,

(4.16) E{ exp(kh(x,(t*),t*))} < exp{C(N,k)T},

where

(4.17) C(N,k) = ?kN-1 W + jk2N-1GH2 exp (kN-'c).

Proof. Let #(x, t) exp (kh(x, t)), and, for any v e [0, T] define v' = min (v, t*). From Lemma 2 and the optional stopping theorem, since h(x(O), O) = 0,


E{exp (kh(xN(v'), v'))} = E FQ(x(uu),du

+ 1 (4.1 8)

E IQ(x(u(),')Iu'du + 1,

where, defining

(4.19) Ah'(x, t) = h(x + JN-', t)- h(x, t),

Q0 is given by

(4.20) Qo(x, t) = exp(kh(x, t)) (kah(x, t)/at

+ N g'(x) [exp(kAh'(x,t)) - 1]}.

By Taylor's theorem, for (x, t) e U,

(4.21) 1 Ah'(x, t)j N-'Hj J I;

Ah'(x, t) - N-'J.Vh(x, t) < 1 N- 2 W;

and so the term in braces in the R.H.S. of (4.20) can be written as

k[Dh(x,t)/1t + ,

g'(x)J.Vh(x,t)] (4.22)

+ ?kN-'•OW+ lk2N-'GHO2 exp(kN - 'c),

where 01 and 02 denote quantities of modulus not exceeding unity, and where the term in square brackets is zero because h satisfies Equation (3.1). Thus (4.18) can be reduced to

E{exp(kh(xN(v'), v'))} < 1 + C(N, k)E{exp(kh(xN(u'), u'))}du,

and by Gronwall's inequality this implies that

E{exp (kh(x,(v'), v'))) exp {C(N, k)v}.

Taking v = T, the Lemma follows, because T' = t* a.s.

Corollary 1. Under the assumptions and definitions of Lemma 5,

(4.23) P[I h(x N(i), i) 1 y] < 2 exp {- ky + TC(N, k)}.

Proof. It is immediate from Lemma 5, taking t* = i, that

P[h(xN(), i),) > y] < exp { - ky + TC(N, k)},

and the Corollary follows because - h is also a solution of(3.1), and has the same bounds (4.15) as h.


The choice of k is arbitrary, and for the purposes of Corollary 1 may be chosen

depending on y and N to give a convenient estimate in (4.23). For example, if k is

taken to be NM, Equation (4.23) implies that

(4.24) P[I h(x,(f), i) I y y] 5 2 exp { - yN1+ ? TGH2(1 + O(N-?))}.

It may in practice be easier to work with this simple estimate than the better one

given by the following corollary.

Corollary 2. Under the assumptions and definitions of Lemma 5, let

a = jWT and b = ?GH2T,

and let Y denote y - aN-'. Then, for y >

aN-,

P[I h(xN(), )] =

y] 5 2 exp{- Nyu(y)(1 + cu(s))/(2 + cu(f))} (4.25)

( 2exp { - -N(y - aN-')u(y - aN-')),

where u(w) is the unique non-negative root u of the equation

(4.26) ub(2 + cu)ecu = w.

Remark. Equation (4.26) implies that u(s) ? ./2b,

with asymptotic equality as Y -* 0: since the significant order of y turns out to be N- *, this estimate of u is

extremely useful.

Proof. Equation (4.25) is deduced directly from Equation (4.23) by choosing k = k(y,N) to minimise its R.H.S.

Lemma 5 can be used to provide information about the tails of the distribution

of x(t*) - .(t*) in the same way as was used after Lemma 4. It does not require the full strength of Assumption B, because it only involves the existence of continuous derivatives of h up to the second order. By Hartman (1964), Chapter V, Theorem 4.1, if the components of ll(x, t) r- (x, t, 0) - x(O) are taken for h, these derivatives exist and are continuous, if the functions gJ(x) are all twice continuously differentiable. Lemma 5 can in fact be further strengthened, by requiring only the

existence of continuous first derivatives for gJ(x). Under this hypothesis, the second

estimate in (4.21) is changed, with a corresponding modification of C(N, k), to N - fJ(N) wherefJ(N) = o(l) as N - co.

Note that no use is made of Theorem K in Section 4: however, Lemma 2

corresponds closely to Proposition (2.1) of [K].

5. Examples

(a) The linear birth and death process Let ZN(t): 0 ? t

< T be the continuous parameter Markov process on the non-

negative integers, with conservative Q-matrix specified by the transition rates

j = 0, 1,2, ..., q

- 1_, = jb,

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and with initial state ZN(O) = Nx0. Define the sequence of processes {xN(t): 0 < t < T} with state space Lk by

xN(t) = N- 1Z,(t), 0 t ?< T.

Then xN(O) = x0, fixed independent of N, and xN(t) is a continuous parameter Markov process of the form assumed in Equation (2.1), with f = { - 1, 1} and

g+l(x) = ax, g-'(x) = bx.

The deterministic Equations (2.2) for this process are

(t) = (a - b)?(t), O < t < T; ?(0) = xo,

and Assumption A is satisfied with

?(t) = x0oexp((a - b)t), 0 _

t ? T.

The set S', for any fixed e > 0, is then the closed interval [xomI -

e, x0om2 + ], where we define

m, = min {1,e(a-b)T, m2 = max {, e(a-b)T},

and Assumption B is clearly satisfied for any e, 0 < e < xom1.

Equation (3.1) for this sequence of processes reduces to

Oh Oh + (a - b)x- = 0, at dx

and it has an integral h(x, t) = xe-(a-b)t: hence, applying Lemma 1, we deduce that WN(t) N+(N-iZN(t)e-(a-b)t - x), considered as a random eln0.t of D[0, T], converges weakly in the Skorokhod topology to the Gaussian random function W with

EW(t) = 0; E{W(s)W(t)} = :2(s), O0 s t < T;

where, by Equation (3.5),

2(s) 2(u = x l'(a + b)(1 - e-(a"-b)s)(a - b)-1, a b,

O x-o1 (a + b)s, a = b.

Alternatively, if we define, for 0 _

u - 1,

W?(u) /(a - b) (xo'(a + b)(1 - e-(a-b)T)- y+

x WN( - (a - b)-'log {1 - u[1- e(a-b)T]}),

when a b, and

Wk(u) = {Txo'(a

+ b)} -W,(uT)


when a = b, then WN, considered as a random element of D[O, 1], converges weakly to standard Wiener measure as N - oo.

The rate of convergence implied by Theorem 1 for the distribution of WN(t) is of order N-*logN, and the bound, if calculated, for Equation (3.8) is found meaningful, for values of T around L-1, only when N exceeds 5000. This should be compared with the results obtained by regarding the ZN(t) process as a sum of N independent processes with initial population 1, and using the Berry-Esseen theorem, giving the expected convergence rate of N--, and a bound meaningful when N exceeds 100. It seems reasonable to suppose that, in other situations also, the approximation will be better than Theorem 1 suggests.

A feature of this particular example is that, for h(x, t) = x exp(-(a - b)t), h(xN(t), t) is actually a martingale, and not just an approximation to one. Similarly, in the notation of Section 4, the sequences {0s} corresponding to •o(x, t) = {h(x,t) - xo}' have the property that

0,s(x, t) = 0 identically in (x, t) for all

s > r, and that ,-.1(xN(t), t) is a martingale. In n-dimensions similar results hold. Suppose that the functions gJ(x) are

linear in x in any region S s: R". Then the deterministic equations (4.3) are of the form

(5.1) j = Ax + b, x S,

for some matrix A and vector b. It is easily checked that, if b(x,t) - x . c(t) + d(t),

where + cA = 0; d+ c.b = 0;

then Q4(x, t) = 0 identically in (x, t)eS x [0, oo): and it can be shown that

{W(xN(i) 0),F, 0 t < co} is a martingale.

By solving (5.1), it can be seen that the deterministic solution rl(x, t u), as defined following Equation (4.5), is linear in x. Let 40(x, t) be any integral of (5.1) which is multinomial of degree d in the coordinates xj, and construct the sequence {•O} corresponding to it, using the formula (4.4). From Equation (4.5), and the fact that g'(x) is linear in x,

L2(•b)(x, t) is of degree at most d - 1 in x, for d

> 2, and is zero if d = 0 or d = 1: hence, from Equation (4.4), 4,(x, t) is of degree at most d - 1 in x for d > 2, and is identically zero if d = 0 or d = 1. An inductive argument based on the same considerations shows that 0,(x, t) is of degree at most d - r for r ? d - 1, and is identically zero if r ? d.

It is then easy to verify that Qd_, (x, t) is zero identically in (x, t)e S x [0, oo), and it can be shown that {tfrd-1(xN(), , -_ 0 < t < oo} is a martingale. Con- venient integrals of (5.1) to take for •o(x, t) are simple multinomial expressions in the coordinates of the vector rl(x, t) - i(x, t, 0) - x(0).

The importance of the linear case is that it can often be used as a local approximation to more complicated systems. The simple form of these exact martingales is one of the advantages thereby gained.


(b) The quadratic birth and death process

The linear birth and death process was taken as an example because its distributions can be determined by other means for the purpose of comparison. The importance of the method is the wide class of processes to which it is applicable, and such a simple example is unrepresentative of its power. The following example, differing only slightly from the previous one, serves to illustrate its use for a sequence of non-regular processes.

Let ZN(t): 0 ? t < TN-' be a continuous parameter Markov process on the

non-negative integers, with conservative Q-matrix specified by the transition rates

qj j+1 = j2a, j = O, 1,2,

..., qj,j-1 = j2b,

with initial state ZN(O) = Nxo, and with a > b. Let

XN(t) = N-1ZN(tN-1'), 0 <

t < T;

then xN(t) is a process of the form assumed in Equation (2.1). Proceeding as in the previous example, we are led to conclude that, if T< [xo(a - b)]-', then

WN(t) _ N){(xN(t))- - xo1 + (a - b)t),

considered as a random element of D[O, T], converges weakly to the Gaussian random function W with

E W(t) = 0;

E{W(s)W(t)} = (a + b){1 - [1 - sxo(a - b)]3} /{3x3(a - b)};

O<s t< T.

However, at t = {xo(a - b)} )-', the solution of the deterministic equations

Z(t) = (a - b)?2(t); ?(0) = xo,

becomes infinite, and so, if T> {xo(a - b)}-1, Assumption A is violated, and the theorems are no longer applicable.

(c) The closed epidemic

Let X_(t) - (X~'(t), X()(t)): 0 < t ? T be a continuous parameter bivariate Markov process on Z+ x Z+, with conservative Q-matrix specified by the transition rates

(j, k)-+(j - 1, k + 1) at rate N-ojk,

(j, k) -+ (j, k - 1) at rate flk,

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and with initial state (X (t), X12)(t)) = (N, Nh), where c, fl and h are fixed

independent of N. This process is commonly used to model the spread of an infection in a closed population, in which the number of susceptibles is denoted by X(') and the number of infectious persons by X(2), the remainder of the population being presumed immune or dead. The first transition then represents the spread of the infection by homogeneous mixing of infectious and susceptible people, and the second transition represents the rate at which infectious people are removed from the mixing population. The dependence of the rates of these transitions upon N, which is proportional to the total population size, is here chosen in its most natural form: however, the initial number of infectious persons is not chosen, as would be natural, to be unity, since this necessitates a further

special approximation during the initial phase of the process. For two generalisa- tions of this model, see Downton (1968) and Ridler-Rowe (1964).

Let xN(t) = N-IXN(t), 0 < t ? T: then the sequence {xN(t)} of processes is of the form assumed in (2.1), with f = {(- 1, 1); (0,- 1)} and

g(- )(x) = cx(~)x(2), g'-1(x)= p(2)

The deterministic equations (2.2) for this process are

d•"()/dt = - 5(1)((2)

(5.2) d?(2)/dt = (c(1)'-flp) ,2)

to be satisfied in some finite interval 0 ? t ? T with the initial condition 4(0) = (1,h). They have a unique solution 4(t), 0 < t < T, for any finite T> 0, given implicitly by

(5.3) '(1)(t) + ?(2)(t)_-logll)(t) -1 -h = 0,

O

(5.4) (1 + h + qu - e")-'du -at = 0,

where q = /l/a is the threshold parameter: thus Assumption A is satisfied. As-

sumption B is also satisfied with

0 <e min(he-T, e(1 +h)/r).

The expressions on the L.H.S. of Equations (5.3) and (5.4) correspond to two

independent integrals of the linearised martingale equation (3.1) for the process: they are

h(x, t) = x(1)+ x (2)_- logx(') - 1 - h, (5.5)

h*(x,t)= (x(') + x() - ilogx() + tlu - e")-du -t. log x( )


With a view to Lemma 1, we define a new pair of coordinate random variables for the process,

zN(t) = (1 - !' (t))yY I(t) + y,

2(t) (5.6)

WN(t) = { - [ 41+(t)] (+[t,]1)(t)]-1- 1)I(t)}y?1(t) - I(t)Y (2t),

where

(5.7) I(t) = f (1 + h + qu- eu)-2du.

We then deduce that, as N-+oo (ZN, WN) converges weakly in {D[O, T]}2 to the continuous bivariate Gaussian Markov random function (z, w) specified by

E*{z(t)} = zo; E*{w(t)} = wo; E*{(z(t) - zo)2

- =

[x-'r2 - ilog x]x=4(t) ,

(5.8) t

E*{(w(t) - o)2= [(1)]-1{[ (2)]-1_

2I I+ l2([q

+ ( 1)]} du,

to E*{z(t)w(t) -

zoWo}-= Pfl - I{(2)[ + ('1)]}du,

for any 0 ? to ? t ? T and any (zo, w0) E R2, where E* denotes the expectation conditional on (z(to), w(to)) = (zo, wo), and where the argument u in the functions ( and 1 within the integrals has been omitted.

As an application of this result we deduce that, as N-+ oo,

P max I zN(t)I>xj m(x, T) =P[ max TB(t)> x], LOt-1T 0J-5 TI 2(T)

where B(t) is standard Brownian motion and where

EI(T) = ,72([(1)(T)]-1_ 1) - _ log ?')(T). This may be rewritten, in terms of the original epidemic process XN, as

P max N Jh(N-'X(t),t) - h(4(t), t) > ]x m(x, T) as N -+o

and may hence be interpreted to give a confidence band around the deterministic trajectory of the epidemic, which the XN process will leave before time T with probability approximating m(x,T). Such information could provide a useful guide to health authorities who, faced with an incipient epidemic, need to predict what emergency resources will be required.


Acknowledgments

The author expresses his thanks to D. G. Kendall, P. Billingsley and G. K. Eagleson for their invaluable stimulus and encouragement in this work.

References BARBOUR, A. D. (1972) The principle of the diffusion of arbitrary constants. J. Appl. Prob.

9, 519-541. CHUNG, K. L. (1968) A Course in Probability Theory. Harcourt, Brace and World, New

York. DOWNToN, F. (1968) The ultimate size of carrier-borne epidemics. Biometrika 55, 277-

289. HARTMAN, P. (1964) Ordinary Differential Equations. Wiley, New York. KURTZ, T. G. (1970) Solutions of ordinary differential equations as limits of pure jump

Markov processes. J. Appl. Prob. 7, 49-58. KURTZ, T. G. (1971) Limit theorems for sequences of jump Markov processes approximating

ordinary differential processes. J. Appl. Prob. 8, 344-356. McNEIL, D. R. AND SCHACH, S. (1973) Central limit analogues for Markov population

processes. J. R. Statist. Soc. B 35, 1-23. RIDLER-ROWE, C. J. (1964) Some two-dimensional Markov processes. Ph. D. thesis, University

of Durham.

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