On the Quasi-Stationary Distribution for Some Randomly Perturbed Transformations of anIntervalAuthor(s): Fima C. Klebaner, Justin Lazar, Ofer ZeitouniSource: The Annals of Applied Probability, Vol. 8, No. 1 (Feb., 1998), pp. 300-315Published by: Institute of Mathematical StatisticsStable URL: http://www.jstor.org/stable/2667247Accessed: 03/06/2010 21:33

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The Annals of Applied Probability 1998, Vol. 8, No. 1, 300-315

ON THE QUASI-STATIONARY DISTRIBUTION FOR SOME RANDOMLY PERTURBED TRANSFORMATIONS

OF AN INTERVAL

BY FIMA C. KLEBANER,1 JUSTIN LAZAR2 AND OFER ZEITOUNI3

University of Melbourne, University of Melbourne and Technion-Israel Institute of Technology

We consider a Markov chain X' obtained by adding small noise to a discrete time dynamical system and study the chain's quasi-stationary distribution (qsd). The dynamics are given by iterating a function f: I -+ I for some interval I when f has finitely many fixed points, some stable and some unstable. We show that under some conditions the quasi-stationary distribution of the chain concentrates around the stable fixed points when ? -- 0. As a corollary, we obtain the result for the case when f has a single attracting cycle and perhaps repelling cycles and fixed points. In this case, the quasi-stationary distribution concentrates on the attracting cycle. The result applies to the model of population dependent branching processes with periodic conditional mean function.

1. Introduction. This paper deals with quasi-stationary distributions for Markov chains X-,

(1) Xs = f(Xs-1) + 68(X81)

obtained by adding small noise (?, which is generally state dependent, to a discrete time dynamical system

(2) Xn+i = f(xn).

The function f maps some interval I into itself, and we consider the case when f has finitely many fixed points, some of which are stable and some unstable. The dynamical system defined by (2) models a particular physical phenomenon, when the variable of interest is confined to the range with set I. For example, in population dynamics xn denotes the population density in the nth generation. The Markov chain defined by (1) models small random perturbations to that phenomenon. Therefore in applications it is often of interest to study the long term behavior of the chain (1) as long as it stays in I. This is done by studying the quasi-stationary distribution (qsd) of the

Received February 1997; revised May 1997. 1Research partially supported by the Australian Research Council. 2Research partially supported by the Australian Postgraduate Research Award. 3Research partially supported by the Australian Research Council and by a US-ISRAEL BSF

grant. AMS 1991 subject classifications. Primary 60J80; secondary 60F10. Key words and phrases. Large deviations, logistic map, branching systems, quasi-stationary

distribution.

300

QSD FOR RANDOMLY PERTURBED TRANSFORMATIONS 301

chain,

(3) p8(A) = lim P(Xn E AI X' E I),

defined for any Borel set A c I. Our goal in this paper is to show that under some conditions the quasi-

stationary distribution of the chain concentrates around the stable fixed points of f when ? -- 0.

Our motivation comes from the population density process in density- dependent branching models. A density-dependent branching process is a branching model where the law of offspring distribution depends on the pop- ulation density. These processes are stochastic analogues of the deterministic models of population dynamics given by (2), and were studied in [6] and [7], where the law of large numbers, the central limit theorem and convergence to a Gaussian process were established. The basic feature of the deterministic models is the decline in the reproduction rate from a constant, when the population density is small, to zero when the population density increases to its threshold. This decline is due to competition between individuals. The logistic model with f(x) = rx(l - x), one of the more famous models (e.g. [10]), provides a perfect example of this phenomenon, and we shall use this model in our representative examples. Density-dependent branching pro- cesses incorporate the same principle, by making the offspring distribution decline with an increase in population density. If m(x) denotes the mean of offspring distribution when the population density is x, then the expected density of the next generation is given by the conditional mean function f(x) = xm(x). The precise definition of the density-dependent branching processes is given in the section on applications; it suffices to say here that these models can be represented in the form of (1) with the dynamics being that of the corresponding deterministic population model with f (x) = xm(x). Under broad assumptions, density-dependent branching processes become extinct with probability 1. We show that the qsd exists and describes the long-term behavior of the process prior to extinction.

In [8], the large deviation principle was established for the Markov chain (1), and this reference is central for this paper. It was shown there that if the deterministic system has a stable periodic orbit then the perturbed system will follow approximately a limit cycle for a time exponentially long in the level of noise before switching to another cycle.

We also consider the model with additive state-independent noise; it may also have an independent interest in applications. This model is conceptually the same, but technically it is easier to treat.

It is not hard to establish the existence of qsd under some broad assump- tions (cf. Theorem 1), the main tool being the Krein-Rutman theory of positive operators. It is also rather straightforward that any weak limit of the qsd as the noise converges to zero must be an invariant measure of the deterministic system, defined by p(A) = p(f 1(A)) (cf. Theorem 2). Therefore, if the deter- ministic system has a unique invariant measure then the qsd concentrates

302 KLEBANER, LAZAR AND ZEITOUNI

around it. We, however, consider the case when the deterministic model may have many invariant measures. The main result of this paper, Theorem 3, is to establish conditions so that the qsd does not put, in the limit of small noise, mass around the unstable fixed points. The technique of proof relies on an adaptation of the discrete time Freidlin-Wentzell theory, for example as developed in [4].

We use the main result to treat the case when f has an attracting stable cycle (an attracting fixed point is a cycle of period 1) to which all the trajecto- ries converge (except unstable cycles and their inverse images) and show that under suitable assumptions the qsd concentrates around the attracting cycle. Thus the limit of the qsd is uniform on the stable periodic orbits.

To put our results into perspective, we point out a general result of Kifer [4], which shows, among other things, that for diffeomorphisms under a bit different conditions of hyperbolic sets and a more special form of random per- turbations, that is, perturbation by a diffusion, the quasi-invariant measures concentrate in the limit on the attractors. While our conclusions might pos- sibly be obtained from general arguments (see [5], Section 5), the case we consider does require some care, as the example provided after the statement of the main Theorem 3 demonstrates.

While working on this paper we learned of a recent, independent work of H6gnas [3], where he obtained similar results for the stochastic Ricker model by techniques not far from ours. The main difference of his work from ours lies in the greater generality and applicability of our assumptions. On the other hand, [3] explains how to extend the assumption (A6) below to allow for an application to the situation of stable periodic orbits of period larger than 2. This extension caries over to the results in this paper; see the comment at the end of Section 2.

2. Assumptions and results. Let I = [0, 1], and consider a continuous, piecewise C2 function f: I -- I. Here (I, X, f) defines a standard dynamical system. For any 8 > 0, let B, = [8, 1 - 8]. Throughout this paper, we make the following assumptions:

(Al) Let f (0) = f (l) = 0, max.,, f (x) < 1, and f possesses a finite number of unstable fixed points x4 = 0, x1, ..., 4 where k may equal zero, and a finite number of stable fixed points s1, ..., s, E (0, 1). Further, for all 80 > 0 small enough, there exist 8'0 > 50 > 0 such that f(B,60) c B,6. [By xi unstable, we mean that If'(x*)l > 1.]

(A2) All trajectories of the dynamical system converge to one of the fixed points of f.

Consider the following Markov chain, which is a random perturbation of the dynamical system. Let

Xn= f(X ) + Xn (Xn-)

QSD FOR RANDOMLY PERTURBED TRANSFORMATIONS 303

Here, 4e(x) are random variables, whose law, denoted Q8, x(1), depends on 8, x only. Let A((x, A) = log(EeA'(x)) and define the kernel P. on (I, M(I)) by

P,(X, A) = Px(X- E A), x E I, A E M(I)

We make the following assumption on Q(, x( ).

(A3) For all e > 0, and for some nonzero probability measure V, on (I, X/3(I)), there exists a nonnegative function mj(x, y) on I x I such that for all x E I and for all A E M(I),

P,(X, A) = ' m, (x, y)V,(dy).

Further, there exist integers no(e) and real numbers a(?) and b(e) such that for all x, y E I,

o < a(?) < mo(no())(X, Y) < b(e) < oc,

where for any n, me (x, y) is the density of Pn(x,.) with respect to V,.

As- sume further that for all 8 > 0 small enough, for all ? > 0 small enough, V,(B,) > 0.

(A4) There exists a AO > 0 such that, for all JAI < AO, supXE,l? A((x, A/c) < oc.

Note that by Chebyshev's inequality, (A4) implies that g8(x) converge in probability to zero as 8 -+ 0 uniformly in x E I.

Let U(x) = (x-8, x + ? ) n [o, 1].

(A5) There exists a ,3 > 0 independent of 8 such that for all 8 small enough, and all 8 small,

XeUno U(xQ X(' 0)) >

inf Qe X(-oo, -8) > /8. XEJ o U5(xW)

(A6) For each i = 1, ..., k there exists an mi > 21f'(x*)J such that for all 5i > 0 small enough, with Ai = Umj5j(X)\U25(Xj), there exists a j(i) < oo

such that f (i)(Ai) C U>=1 U8/2(si)

For i = 0 we set mo = 31f'(O). For each i = 0, ..., k, define

-r 1 = min{n: XA JiXi i'l n fl U2i X

and

i, 2= min{n: XA E Ua6 (xi)

In what follows, X denotes the weak convergence.

304 KLEBANER, LAZAR AND ZEITOUNI

THEOREM 1. Assume (A1)-(A4).

(i) The qsd p(?) [cf. (3)] exists; it is a probability measure on (I, A(I)) and there exists a number R(?) > 1 such that for all A E O(I),

P(?)(A) = R(-) f P,(X, A)p(8)(dx).

(ii) lim8 0 R(8) = 1.

Lt(1) (n) X Let m8 (x, y) = m8(x, y) and M. (x, y) be the density of pn for n > 2, that is, Pn (x, dy) = (71) (x, y) dy.

PROOF. (i) By (A3) and Theorem 10.1 of [2], P8 has a positive eigenvalue 1/R(I), larger in magnitude than any other eigenvalue, and a corresponding uniquely defined left eigenfunction p(?) satisfying fI p(?)(y) dy = 1. Moreover, still by [2], Theorem 10.1, with (?)(Q) denoting the (uniquely defined) corre- sponding right eigenfunction,

(4) m)( x, y) - ( R) ) R u) (x)W(8)(y)[1l + o(1)],

where the convergence of the error term in (4) is uniform in x, y. Then for A E 0/3(I), define p(8)(A) = fA p(?)(y) dy, so that it follows directly that p(?) is a qsd by using (4) in

P(X' E AIX- E I)= P(XnEA) n n ~~P(X-o E I

(ii) With 8o, 8'0 as in (Al), it holds that for any x E B,80,

Px(X" E Bc ) < Q? x(-o, 80 - 8') + Q?, X(- O O )

< exp(-A0180 - 8'I/e)(exp(A1(x, AO/e)) + exp(AV(x, -A0/e))).

Hence, by using (A4), for some constant c > 0,

(5) sup Px(X- E B,c) < ec/

Hence,

a. -SUp (1-Px(X- E B,6o)) -- ?' 0.

Then since p(-)(B,60) > 0 for all ?, p(e)(B,60) = R(e) f Px(X- E B80)p( )(dx) > R( f) (1 - a,)p(8)(dx)

I B8~~~~~~~~~~~~~~0 implies

(6) R(? < I 0, 1-a?

and since R(e) > 1, (ii) follows. ri

QSD FOR RANDOMLY PERTURBED TRANSFORMATIONS 305

Since I is bounded, p(?) is tight. The following identifies the limit points of subsequences of p(?).

THEOREM 2. Assume (Al)-(A4). Then, any weak limit p of p(?) is an invari- ant measure for f, that is, for any A E M(I),

p(A) = p(f (A)).

REMARK 1. Kifer ([41, page 8) proves this result in the case of stationary distributions of Markov chains under somewhat different assumptions that are not satisfied here.

PROOF OF THEOREM 2. Let Y8 and Y be random variables with distribu- tions p(?) and p, respectively, with Y? # -0 Y. For any A E M(I),

(7) P(Y8 E A)= R(6) f P,(X- e A)p(-)(dx) = R(-)P(f(Y,) + 4e(Y,)) E A).

By continuity of f and (A4) [see the comment on (A4)], f(Y8) + ge(Yj) X

f(Y) along the subsequence. Hence for all A E M(I) such that P(Y E dA) =

P(f(Y) E dA) 0 O, taking ? to zero along the subsequence in (7) along with part (ii) of Theorem 1, gives

(8) p(A) = p(f 1(A)).

Hence we can now show that (8) is true for any interval in I by approx- imating the interval where necessary by sets that do satisfy it. Hence, by Caratheodory's extension theorem, it extends to all A E 2(I0). Now, with 0 < 8 < 1 - maxXEI f(x) fixed,

p(e)((1 - 8N, 1]) = R() | PX(X1 E (1 - 8, l])p(-)(dx)

< R(-) max P(g:(x) > 1-8-max f(x)) - 0 as ? - 0 xel xel

and so p({l}) = 0. Hence (8) is true for A = {0} and A = {1}, and so it is true for all A E M(I). D

It is rather straightforward to check that any weak limit of p(?) then satisfies

Ei=o p(xi) + E> p(s*) = 1. The following is our main result.

THEOREM 3. Assume (A1)-(A6). Then, any weak limit p of p(?) satisfies p({xJ}) = 0 for each j = O,..., k.

REMARK 2. To see that one needs some structural assumptions on consider the case of f(x) = rx(l - x), 1 < r < 3. For x E [e, 1] let (?(x) be a uniform random variable on (-?/2, ?/2), while for x E [0, ?) let the law of (?(x) be c,U(-rx, -(r - 1)x) + (1 - c,)U(1 - e/2r, 1], where U(A) denotes the uniform law on A and C 8 >? 1 is chosen such that E48(x) = 0. It is not hard to check that in this case, the qsd concentrates on the (unstable) point 0.

306 KLEBANER, LAZAR AND ZEITOUNI

To prove Theorem 3 we need the following preliminaries. Throughout, we use the notations introduced in (A1)-(A6). In addition, N := N(e) = K(log 1/8)K for some K large enough (and independent of ?).

LEMMA 1. For each i = 1, ..., k,

(9) sup Px XJ(j) E (U Ui((Sj))) 0

PROOF. By (A4), supXEI P(((X))I > c) -- 0. This implies that for all 8 > 0, SUPXEI PX ( X X1-f (x) I > 8) > ?0 0. Iterating this and using the fact that j(i) is independent of ?, the lemma follows. El

LEMMA 2 ([8], Lemma 2.1). For each i = 1, ... , 1, for each j = 1 ..., 1 for all k > 0,

sup P (r 2 <kie) le 0? XEU5i (Si)

LEMMA 3. For each i = 1, ..., k, for all K > O,

/ k

sup sup p E U8(4)) ?0 ? xEAi j(i)<t<K(log(1/e))K m=O

PROOF. Omit the i subscript for convenience. Take j as in (A6). Then for all x E A, using the Markov property,

sup Px(Xt E U8(x*)) j<t<N

< sup PX(Xy+t E U8(x*)) t<N

(10) k < SUpsuUpsup Py (X U U,, (X))

YEUm=1 U5(Sm) t<N m=O E

+ PX ( E (u UA S771 ))

< max sup Px(T' < N) 1<m<l XEU2(Sm)

(11)~ ~ ~ ~ ~~~ + PX(XJ E (U U6(Sm))) 0 as - 0,

where the last limit is due to Lemma 2 and Lemma 1. El

LEMMA 4. For each i = 0, . . ., k, we have the following:

()SUPXEU,i(Xi*) PX(X'il ?1 UMi5i (X*i)) *10;

(ii) for all K large enough, suPXEU8 (X*) PX('ri 1 > K(log(1/e))K) 0 as ? - 0.

QSD FOR RANDOMLY PERTURBED TRANSFORMATIONS 307

PROOF. Omit the i subscript for convenience. (i) By the choice of m in (A6) [in0 is defined immediately after (A6)], we

have that for all 8 small enough,

(12) sup lf(x)-x*l<m5 XEU28(x*)

and so there exists a constant A > 0 such that

sup Px(X-1 ?' Un?8(x*)) sup PX(f(X8'_1) + f (X-?1) UMAX

xeU5(x*) xeU5(x*)

(13) < SUP PX( ~(Xe) _ Ap) XEU5(x*)

<supP(O ?(x) >A)-O0 as -O0. xel

(ii) Let be = e(log e)2. Define T = inf{t: I X- - x I > b8}. Without loss of generality, we assume that infxEU8(X*) f'(x) > c> 1. Then, with c1 = 4/ log c,

inf PX(T < c, log log 1/8) x* <x<b,+x*

> inf P((8(x) >) inf Px(ir < C1 log log 1/8 -1). x 8<x<b,+x*

Note that as long as xi E U,(x*), on the event {f (xi) > 0}, f(xi)+f?(xi) > cxi. Hence, since

8CCl log log 1/e = 8(log ?)4 > b,,

one gets, using (A5),

-cl log log ? inf Px (' < c, log log 1/?) > (inf P'((W(x) > ?

x*<x<b,+x* xeU\(x*)

> j3-ci loglog?

Arguing similarly, the case x < x* is handled (for i 7& 0). Hence,

inf PX(T < clloglogl/8) > j8-C log log lx*-xl<b,

It follows that, with c2 = 2c1,

inf PX (T >C C13C2 log log, log log 1/8)

(14) lx*-xlb,

< (I1 f-,Clogjog0),8C21091098 as 8 - 0.

308 KLEBANER, LAZAR AND ZEITOUNI

In what follows, we use C3, C4, C5 to denote some constants which are indepen- dent of 8. Then, with gr = 4(X ), arguing as above,

x>b,+x*,xeU8(x*) x

(bZ,

1 - lo a)

> P(i < b8/,v/c_ i = 1 1logl/b,)) (Tel log( I

og fc

(15) l-~,)~og b, I ogv,/ (15) ~~> (1_ C3)

> exp (7b log b) > exp (7) 1 as E 0.

Combining (14) and (15), we conclude that for some c5 independent of 8,

inf Px(Ti 1 > (log 1/?)-C510) -> 0 as -> 0. xeU,5(x*)

This concludes the proof of the lemma. D-

LEMMA 5. For any i = 1,..., k, k

suP PX XN E U U8 (X)) ? as E 0. xeU8i (x*) i =

PROOF. For x E U8. (x7), using the Markov property,

PX (XN E U U, PX(T1 > N) + Px(X-Z1 ' Umi1 (x ))

(16) k

+sup sup P X( E U U,6 xeAi l<t<N j=o

Taking supremum on both sides over x E U8. (x*), the first two terms on the right converge to zero as 8 converges to zero by Lemma 4, while the convergence of the third is a consequence of Lemma 3. D-

LEMMA 6. For any 80 small enough,

sup PX(X r E [0,8 0)) -> 0 as E > . xe[O, 80]

PROOF. Note that

sup PX(XN E [0, s0)) < sup PX(TO 1 > N) xe[ 8O 50] xe[O, 80]

(17) +E> sup Px(X701 ' [2280, 31f'(0)180])

+ N sup PX(X1 E [0, 50])* xE[80, 1-80]

QSD FOR RANDOMLY PERTURBED TRANSFORMATIONS 309

The first two terms in (17) converge to 0 with 8 due to Lemma 4, while the third one is bounded [due to (5)] by Nec/1 -+ 0 as 8 - 0. C]

PROOF OF THEOREM 3. Fix any j = 0, ..., k, and pick 8j small enough so that

p(-)(U6 (X)) -> p({XD}) as 8 -? 0.

First observe that R(s) < 1 + e-c/?, giving

(R(--))N 1-> as s?-- O7

for K large enough. Now,

P(-)(U,6 (X*-)) (R(E))N f

P(XN E U. (X ))p(s)(dx)

- (R(8)) E>U11 * PX(XN E Ujx)p(X

(18) + (RN) E I )PX(XN E U8(XJ))p(8)(dx)

? (R(s))N B PX(XN E U8.(X ))p(S)(dx)

+ (R(8))N f PX(XN E Uj( (-)(dx)

U~~0 (Nxx*))

where B = [O, 1]\{JUk U8.(x*) U>=1 U8(si)}, so that the third term converges to zero as e converges to zero, the first term converges to zero as e converges to zero using Lemma 5, and for the second term, for each i = 1, ..., 1,

IUN(si x(N E i(x*))p(?(X

(19) <~ sup PX(XN E Uj6

< sup PX(,Tj2 < N) 0 as e O, xEUSi (Sie)

by Lemma 2. For j = 0, the last term converges to 0 with s as a consequence of Lemma 6, implying that p(-)(U60(x*)) 0> as 8 -> 0. Substituting again in (18) for any j 7& 0, this is enough to imply that p(-+)(U bj(X*)) 0 as 8 -? 0. D

REMARK 3. We comment here on extensions of Theorem 3 which follow from [3]. As pointed out in Section 3, assumption (A6) is hard to check in general. It is used in the proof, however, only to show that points in the neigh- borhood of unstable fixed points converge in finite time (under the determin- istic action) to neighborhoods of stable fixed points. This assumption can be replaced by the assumption of nonexistence of heteroclinic orbits, which in-

310 KLEBANER, LAZAR AND ZEITOUNI

clude the unstable fixed point. This assumption creates an ordering of the unstable fixed points in such a way that neighborhoods of higher order unsta- ble points are mapped to either stable points or neighborhoods of lower order fixed points, and the lowest order unstable fixed points do satisfy (A6). See [3] for a development of this approach in a particular case.

3. Applications and examples.

The logistic map with normal noise. Consider the case when f is the lo- gistic map, f(x) = rx(l-x), 1 < r < 1 + 6:

(20) Xs = f(Xs1) + 7

where f as above and ijn are iid standard Normal rv's.

(A). Consider first the case when 1 < r < 3. In this case x4 = 0 is the only unstable fixed point, s1 = 1 - l/r is the only stable fixed point and there are no other cycles. Assumptions (Al) and (A2) are satisfied for this case; see [11] and, as k = 0, (A6) is not required for this case. If (?(x) = 871, where -1 is the standard Normal variable, then the assumptions (A3), (A4) and (A5) are clearly fulfilled, and the result holds, that is, the qsd of Xn concentrates around the attracting fixed point sl.

REMARK 4. In this case it is possible to show that p(O) = 0 by using a Lyapunov function approach; see [9].

(B). Consider the case 3 < r < 1 + 6.

In this case f has two unstable fixed points at zero and 1 - l/r, one stable attracting cycle of period two, and no other cycles. We next show that Theorem 3 applies in this set-up, implying that the qsd of the X- in (20) concentrates on the stable attracting cycle.

Define the function f*(x) = f2(x), X E I. We shall apply the above results by replacing f with f *. It is easy to see that f * has two unstable fixed points at x* = 0 and xl = 1 - l/r, two stable fixed points s1 and S2, and no other cycles.

First we verify assumption (A6) for f *. We start with a lemma.

LEMMA 7. Let f be the logistic map with 3 < r < 1 + 6. Then the accu- mulation points of f-n(x*) are zero and one.

PROOF. Here f maps [0, 1] onto [0, r/4], and r/4 < 1! f from [0, 1/2] -

[0, r/4] is one-to-one. Put g(x) to be the inverse map from [0, r/4] onto [0, 1/2],

1/ x g(x) = - -

2 V4r

QSD FOR RANDOMLY PERTURBED TRANSFORMATIONS 311

and note that it is monotone increasing. Then, f'(x*) = {xt, g(x*)} = {xt, 1- x* }. We prove that

(21) 1-g(2)(x*) > r/4,

hence the "other inverse branch" of f does not contribute, and all further inverse images of x* come from g. Inequality (21) follows from

(22) g(x*) < f(r/4).

The monotonicity of x(1 - x) implies that g(x*(r)) is monotone decreasing in r for any r > 2. Therefore max3<r.<1+6g(x*(r)) = 1/3 is achieved at r = 3; f(r/4) is monotone decreasing in r on (3/4, (1 + 6)74). Therefore

min3<r<1+06 f (r/4) is achieved at (1+ 6)74) and is f ((1+J6)74) = 0.409 > 0.4. The inequality (22) now follows, hence (21) is established. It follows by symmetry of f that the accumulation points of f -(x*) are the accumula- tion points of {gn(x*)} and of {1 - gn(x*)}. Since g is monotone increasing, {gn(x*)} form a decreasing sequence. The limit y must satisfy y = g(y), which implies that y = 0, and the lemma is proved. D-

It is known that the iterates of all the points in I except the unstable fixed points and their inverse images for all n = 0, 1, 2, .. ., converge to the stable cycle; see [11], page 73.

Since f-n(x*) {0, 1} for all n, it now follows from Lemma 7 that for all 8 > 0 small enough, for all x E U6(X X

f12(x) -{s1,s2}, n -y oc.

With A1 as in the statement of assumption (A6), we have that for each x E A1 there exists an n(x) such that (f*)i(x) E U2i=1 U8,12(si) for all j > n(x). Using the continuity of f, there exists thus a r(x) > 0 such that

2

V Y E Br(x)(x), V j > n(x) + 1, (f*)j(y) E U U812(Si). i=1

Covering A1 by the balls B,1(X)(x) and taking a finite subset by compactness, (A6) follows for f *.

Consider Y- = X.2n It is easily verified that assumption (A3) holds for (20), and that by Theorem 1, the qsd of X- exists. It is clear that X' and X- have the same qsd. Let

4,(x) = f (f (x) + --nl) + 822 - f2(x).

With this choice of (?(x) we can write that

2(n+1) = f2(X2n) + 4 (X21)-

312 KLEBANER, LAZAR AND ZEITOUNI

To verify assumptions (A4) and (A5), notice that the iterated noise satisfies, by the mean value theorem,

4 (x) = f (f (x) + 82nl) + 8212 -f (2)(x)

= 822 + f'(O)821

= 8(22 + f'(0)71),

where 0 E (f (x), f(x)?+ 811) can be random. Since the derivative f' is bounded on I, -r < f'(x) < r, If'(0)l < r. Using this, (A4) is straightforward. To verify (A5), write with f3 = P( 2 > 1 + r 1 1J),

inf P(4'(x) > 8) = inf P(712 + f'(0)j1 > 1) Us(x*) Us(x*)

> uin(f P(72, >1 + rl 1l) = 8 > O. U,5(x*)

The other side of (A5) is similar. This completes checking (A1)-(A6). Theorem 3 implies than the limit of the qsd is of the form a8S, + (1 - a)8s2,

where 0 < a < 1 and Qsi denotes the point mass at si, i = 1, 2. Since Sl, S2 are points of the cycle for f, it follows that a = 1/2, that is, the limit is the uniform distribution on the stable cycle.

REMARK 5. The following is useful for checking assumptions (Al) and (A2). The dynamical system defined by iterations of f E C0 is called simple if for any starting point the trajectory converges to one of the cycles of f. Then f is said to belong to class G21- if it has a cycle of period 2m and no cycles of period greater than 2m. The result [11], page 73, states that if f E G2n, for some m, then the dynamical system defined by iterations of f starting from any point in I is a simple dynamical system. On the other hand if f is C1 and if the dynamical system defined by iterations of f starting from any point in I is a simple dynamical system, then f E G2nz, Theorem 3.3 in [11].

4. Density-dependent branching processes. In this section we apply the main result to the model of density-dependent branching processes. Let {X-} be the population density in a density-dependent branching process, de- fined as follows. For any fixed x > 0, let Yj n(x) be independent and identically distributed random variables for all j, n with distribution Y(x), where for all x > 0, Y(x) is nonnegative and integer valued, and for all x > 1, Y(x) = 0. Here Y(x) represents the law of offspring distribution when the population density is x. Then for fixed K E [2, oo) define a population density-dependent branching process {ZK}, n = 0, 1, 2, ... with threshold K inductively by tak- ing zK to be a positive integer less than K and

Zn+ = |n vnl ZnK > O zn+1 j=

QSD FOR RANDOMLY PERTURBED TRANSFORMATIONS 313

where we assume that for any fixed x, K and n + 1, that the Yj 11+1(x), j = 1, 2, .. ., are independent of ZK, ZK17 ..., ZK. Let XK = ZK/K denote the population density.

For this application it is suitable to take I = (0, 1) to be the open interval. Then if XK E I, with f (x) = xEY(x) = xm(x),

1KX,,

n+= f(Xn ) + (Yj,n+l(Xn) -m(X))

(23) 1

= f(XK) + _Z

where Y'(x) = Y(x) - m(x) denotes the centered offspring distribution, and xK 1 = O if XK V J.

To apply the main result to the chain {XK}, set E = 1/ K, and put

1 Kx x/?2

(4(?x= E Y>x) =2 E Y9 (x). K j=1 j=1

Taking f (x) to be the logistic map rx(1 - x) with 1 < r < 1 + 6, the assump- tions (Al), (A2) and (A6) on f(x) were verified in the previous section on the logistic map with additive noise.

To verify the rest of the main assumptions, we make the following assump- tions on the offspring distribution Y(x).

Assume (A3) with V. being a discrete measure on i/K, i = 1 .., K- 1. Further, let Ay'(x, A) = log E exp(AY'(x)), and assume that

(BI) sup (x/ 2)A '(x, AE) < co. XEI, ?

To show (A4), write, by using independence,

X/?2 2/

\((x, A/?) = log E exp AE YJ = log E ((X)) logExp(~8 x/2 ) (f{ exp (A EY'J(x) j=1 =

X/? 2

= log H1 E exp(A8Y'3(x)) = x/?2 log E exp(AEY'(x)) j=1

= X/2A'(x, AE).

Thus (Bi) implies (A4). To give a specific example, take Y(x) to be a Poisson(m(x)) random variable

where f (x) = xm(x). For the logistic dynamics, m(x) = r(l - x) for x E I and zero for x V I. Then Ay'(x, A) = m(x)(eA - 1) - Am(x). So that

X/? 2A Y(A) = f (x)(eA, - 1 - A?)/?2.

Using expansion for the exponential, (Bi) clearly holds. (A3) also holds.

314 KLEBANER, LAZAR AND ZEITOUNI

Before we verify (A5) we prove a result on sums of iid rv's:

1 Kx X/?2 ?(x)

>E Y' (X) = 8 L(Y (X)- m(x)) =

Let v(x) = E(lY'(x)13) and (-2(X) = Var(Y(x)) = E(Y'2(x)).

LEMMA 8. Let A c I be such that

(25) sup V(Y) Y xEA 0-3(X) -Ix-

Then sums (1/ K) i Y'3(x) converge in distribution as K X to the N(O, Xo-2(x)) distribution uniformly in x E A.

PROOF. Indeed, fix x, and using the Berry-Esseen inequality ([1], page 542), we have for any real a, with 1D being the standard Normal cdf,

(26) P YJ'(X) < a() x) )-x )(a) < 9- (x) k. K _ - o-3(X)NIKX

Replacing a by a/(o-(x) x), we obtain convergence to the N(O, xo-2(x)) dis- tribution; moreover, the convergence is uniform in x if condition (25) holds. Dz

An easily checked sufficient condition for (25) is (B2).

(B2). Assume that there is a small y > 0, such that with A = (I, 1- supxEA v(x) < oo and infX,A cr2(x) > 0.

It is clear that (B2) implies (25). Now the uniform convergence of the sums to the normal distribution implies (A5). Indeed, as it was assumed that unstable fixed points x*, i = 1, 2, ..., k are in the interior of I, there is y > 0 so that A includes all of these points. Now,

inf Q~ x((, 00) > inf P(46(x) > 8) XEJuk1 US(XW) xEA

~inf ( 1Kx nxeA P

>$(Yj(x) -

m(x)) > 1) =, > 0.

(B2) is clearly satisfied for the specific example when Y(x) is a Poisson(m(x)) rv.

This completes the check of the basic assumptions for branching processes when f (x) = xm(x) has a single attracting fixed point, for example, when f is a logistic map and 1 < r < 3. In this case the main result implies that the qsd concentrates around this point. We formulate this as Theorem 4.

THEOREM 4. Suppose the offspring distributions satisfy (Bi) and (B2). Then p(x*) = O.

QSD FOR RANDOMLY PERTURBED TRANSFORMATIONS 315

When 3 < r < 1 + 6, such an attracting fixed point does not exist, but there is an attracting two cycle. The analysis via f(2), similar to that carried out in the additive case and details of which are omitted, shows that the qsd concentrates on the attracting cycle. These results for the stochastic Ricker model were obtained independently by Hognas [3].

REFERENCES

[1] FELLER, W. (1971). An Introduction to Probability Theory and Its Applications 2. Wiley, New York.

[2] HARRIS, T. E. (1989). The Theory of Branching Processes. Dover, New York. [3] HOGNAS, G. (1996). On the quasi-stationary distribution of a stochastic Ricker model.

Stochastic Process. Appl. To appear. [4] KIFER, Y. (1988). Random Perturbations of Dynamical Systems. Birkhauser, Boston. [5] KIFER, Y. (1990). Principal eigenvalues, topological pressure, and stochastic stability of equi-

librium states. Israel J. Math. 70 1-47. [6] KLEBANER, F. C. (1993). Population-dependent branching processes with a threshold.

Stochastic Process. Appl. 46 115-127. [7] KLEBANER, F. C. and NERMAN, 0. (1994). Autoregressive approximation in branching pro-

cesses with a threshold. Stochastic Process Appl. 51 1-7. [8] KLEBANER, F. C. and ZEITOUNI, 0. (1994). The exit problem for a class of density-dependent

branching systems. Ann. Appl. Probab. 4 1188-1205. [9] LAzAR, J. (1996). On the quasi-stationary distribution of randomly perturbed discrete time

dynamical systems. Ph.D. dissertation, Univ. Melbourne. [10] MAY, R. M. (1976). Simple mathematical models with very complicated dynamics. Nature

261 459-467. [11] SHARKOVSKII, A. N., MAISTRENKO, Y. L. and ROMANENKO, E. Y. (1993). Difference Equations

and Their Applications. Kluwer, Dordrecht.

FIMA C. KLEBANER DEPARTMENT OF MATHEMATICS

AND STATISTICS UNIVERSITY OF MELBOURNE PARKVILLE AUSTRALIA 3052 E-MAIL: f.klebaner@ms.unimelb.edu.au

JUSTIN LAZAR DEPARTMENT OF STATISTICS UNIVERSITY OF MELBOURNE PARKVILLE AUSTRALIA 3052 E-MAIL: lazarj@anz.com

OFER ZEITOUNI DEPARTMENT OF ELECTRICAL ENGINEERING TECHNION-ISRAEL INSTITUTE OF TECHNOLOGY HAIFA 32000 ISRAEL E-MAIL: zeitouni@ee.technion.ac.il