On the existence of an invariant probability and the functional central limit theorem of a class of nonlinear autoregressive processes

ON THE EXISTENCE OF AN INVARIANT PROBABILITY AND THE FUNCTIONAL CENTRAL

LIMIT THEOREM OF A CLASS OF NONLINEAR AUTOREGRESSIVE PROCESSES

CHANHOLEE* AND YOUNGMEE KWON

ABSTRACT. Existence of a unique invariant probability is considered for a class of Markov processes which may not be irreducible and a functional central limit theorem for a class of nonlinear irreducible uniformly ergodic processes is derived as well.

1. Introduction

Let (8,p) be a metric space, r a set of measurable maps on 8 into itself, Ta o--field onr such that the map (')',x) ~ ,(x)is measurable on (r x 8,T 08(8)) into (8,8(8)), where8(8) denotes the Borel o--field of 8.

Let P be a probability measure on (1',7). On some probability space (Q,F,Q) is given a sequence of i.i.d. random maps aI, a2, ...with com- mon distribution P. For a given random variableXo,independent ofan, define Xl =alX_{O, ••• ,}X n= _anXn-l = an ... alXO •••• Then, in view of the independence ofan, X nis a Markov process on 8.

Letp(x,dy) denote the transition probability given by (1.1) p(x, B) = P(bEr :,(x)EB}), x E 8, B E 8(8).

We often writeXA^x)for Xn in case Xo= x. Denote by pnthe product measure P x··· x P on(rn,Ti9n). A probability measure 7fon (8,8(8)) is said to be invariant for pif7f(B) = Jp(x, B)7f(dx) , B E 8(8).

Received April 20, 1999. Revised September 30, 1999.

1991 Mathematics Subject Classification: Primary 60GlO, 60J05.

Key words and phrases: invariant probability, functional central limit theorem.

*The author wishes to acknowledge the financial support of the Korea Research Foundation made in the program year of 1997.

We shall writep(n)(x, dy) for the n-step transition probability withp(l) =

p. Thenp(n) (x,dy) is the distribution ofXn(x).

In this article S is a closed subset of JR.I. For r one takes a set of measurable monotone (non-increasing or non-decreasing) functions on 8 into itself. We say that p is <p-irreducible with respect to a nontrivial measure<p onS if<p(B) > 0 implies that for each x there exists n such thatp(n) (x, B) >O.

The transition probabilitypmay not be cp-irreducible for any nonzero^{( j -} finite measure<po One of the two main interests in the paper is to look at one such class of processesXn to get some conditions under which there exist unique invariant probabilities'Jr. And the other is, for an example of such process, which isknown as a nonlinear 1st-order autoregressive process, under irreducibility, to identify broad classes of functions 'I/J in

£2(8,'Jr) for which the functional central limit theorem (FeLT) holds, Le., the sequence of stochastic processes.

(1.2)

n-1/2 [ ~(,p(Xi)^-

J

J

converges weakly to a Brownian motion under the initial distribution'Jr.

Even though the main results in the article are stated and proved for the one-dimensional case, it will be readily turned out that one can extend the results to ]Rk(k ~2) with more relaxed conditions on 8.

2. Existence of a unique invariant probability

Define the transition operatorT on the linear spaceB(8) ofall real- valued bounded measurable functions on 8 by

(2.1) (Th)(x) =!h(y)P(X, dy), hEB(8).

We shall say thatp(x,dy) has the (weak) Feller-property if(Th) is con- tinuous whenever hE B(8).

We make the following assumptions:

(A1) There existsXo and a positive integerno such that Q(Xno(x) ~xo Vx) > 0, Q(Xno(x) ~ Xo Yx) > O.

(A2 ) p has the Feller-property.

Let 8 c JR^{1 ,} being the state space of the process, be a closed interval (not necessarily bounded) and let r be a set of continuous monotone maps , on 8 into 8.

Write

Al {(-00, x] n8 : x ~ xo}, A₂ {(-oo,x]n8: x < xo}, A₃ ([x,oo) n8: x :s; xo},

~ ([x, 00)n8 : x >xo}, A U;=lA.

Then it is simple to check that ,-lA^C A V, E r, outside of a set of P-probability zero. On the spaceP(8), define the distance dby

(2.2) d(J.L,v) =sup{IJ.L(A) - v(A)1 : A EA}, J.L, v EP(8)

The following result may be found in Bhattacharya and Lee (1988) and its correction notes.

LEMMA 2.1. ThespaceP(8)is completeundertile distance d defined by(2.2).

It is easily noticed that the lemma still works for our case also. One of the results to be noted in this article is

THEOREM 2.1. [f(A1 ),(A2 ) bold, then there existsa uniqueinvariant probability^7f for p(x,dy).

For the proof, in despite of differences in situations, one may follow the similar ways found in Bhattacharya and Lee (1988), and here we present a sketch of the proof, including key steps.

First, we note that, by (2.2), the map J.L ^---+ J.L0 , - I is a contraction.

Define the (adjoint) operator T* acting onP(8) by (T*J.L)(B) =

J

It follows that J.L ---+T*J.L is a contraction in metric d. Write r 1 {hI, ... , ,no) E rno;,no ... '1(8) c (-00, xo])

r2 = {hl,"",no)Erno ;'no""1(8)c[xo,00)}

Then

if 'Y Er1 and A^E Al if 'YE r² and AE A2

if 'YE r¹ and AE At

or

if 'Y E r2 and AE A3 where 'Y = ("11, ... ,'Yno)' Therefore, for A E AlUAt,

IT*no(A) - T*nov(A)I

(2.3) ~ ^Jrno 11l(("Ino" ''Y1)-lA) - v(("Ino" ''Y1)-lA)! Pno(d'Y1'" d'Yno)

~ ⁽ !1l(("Ino ... 'Y1t 1A) - v(("Ino'" 'Y1)-1A)\ Pno(d'Y1 '" d'Yno) JrTlfJ\rl

~ (1 - pno(r1»d(ll, v).

For A E A2UA3 , similarly,

IT*no(A) - T*nov(A)I~ (1 - Pno(f₂»d(ll, v).

Combining (2.3), (2.4),

(2.4) d(T*no ll, T*no v )~ ^{max {1-} pno(r1), 1 - pno(r2 )}d(ll, v).

(2.5) and the fact that Il ---+ T*Il is a contraction in metric d together imply

d(Tmll, T*nv )~ 8[n/^no1d(/1, v), '<In =^{1,2, ... ,}

where rn/no] is the integer part ofn/no, and 8= max{1 - pno(r1), ^1- pn^o(r_{2 )},}which is less than 1.

For n' >n, one has

(2.5) d(p(n)(x,dy), p(n')(x,dy» = d(Tmll, T*n v )~ 8[n/no]

with Il =_8x (point mass at x) and v =T*(n'-n)8x'

Hence, p(n)(x,dy) is a Cauchy sequence in the metric d. Let 1T be its limit, which exists by Lemma 2.1.

Then1T is uniform in x and letting n' ---+00 in (2.6), we see that supd(p(n)(x,dy), 1T(dy») _~8[n/no]---+ 0 as n ---+00.

xES

Therefore, p(n)(x,dy) converges weakly to the same probability measure 1T(dy) on Sfor every x E S. Sincephas the Feller property, it is trivial to show that 1Tisinvariant.

If1r' is another invariant probability, then Tm1r,=1r', which implies d(-lr',1l'') = d(T*^n1l",1r) ---. 0 as n ---. 00.

Therefore, d(1r',7r) = 0, so that 7r'= 7r.

3. Some results on the functional central limit theorem un- der irreducibility

As an example of the process X_n considered int he previous sections, which is generated by successive iterations of an i.i.d. sequence of ran- dom maps, on a probability space, consider the following process (3.1)

which is generated by the random maps an defined by x ---. frnX =

f(x) +^Cn (n ~ 1), where f is real-valued Borel measurable and con- tinuous monotone on S, {cn : n ~ 1} is a sequence of LLd. random variables and X_ois arbitrarily prescribable real-valued random variable independent of{cn :n ~ 1}.

With the assumptions (AI) and (A_{2 )} hold, as though there exists a unique invariant for the process, the transition probability p may not be cp-irreducible for any nonzero O"-finite measure cp, not even strongly mixing. Indeed, the tail O"-field may not be nontrivial. Here, the irre- ducibility plays a role, under which it enables us to see these properties hold. A cp-irreducible aperiodic Markov process with transition proba- bilityp(x, dy) is said to be geometrically (Harris) ergodic if there exists a probability measure7r such that

(3.2) I^Ip(n)(x, dy) - 7r(dy)II ---.0 exponentially fast as n ---.00,

\IxE)RI.

Here 11 . 11 denotes the variation norm on the Banach space of finite signed measure on ()RI,B()RI)).

Recently there have been considerable works on kth-order (k ~ 1) nonlinear autoregressive models, most of which provide some verifiable criteria for geometric ergodicity (see, e.g., Chan and Tong (1985), Tj4>stheim (1990), Bhattacharya and Lee (1995a), (1995b), Lee (1998)).

(3.3)

If (3.2) holds, then11"is necessarily the unique invariant probability for p(x, dy), and the process having11"as the initial distribution is stationary.

We assume that the process (3.1) is uniformly (geometrically) ergodic, that is, SUPxES IIp(n)(x, dy) - 11"(dy)II goes to zero exponentially fast as n -+ 00, and that Xohas the unique invariant11"as its distribution. Note that, by contractivity, the convergence automatically has geometric rate.

A set B E B(8) is said to be small(with respect to 'P) if'P(B) > 0, and for every A ^E B(8) with 'P(A) >0 there existsj 2: 1 such that

j

inf '"'p(n)(x,A) > o.

xEBL...

n=l

It also turns out that uniform ergodicity can be characterized as the smallness of the state space 8, that is, a 'P-irreducible aperiodic Markov process with transition probability p(x, dy) is uniformly ergodic if the state space 8 is small.

Let 7j; belong to the range(c L² ==L²(S,11"» of the operatorT - 1on L^2,whereT is the operator defined by (2.1), and I is the identity. Then the functional central limit theorem (FeLT), as stated in the following proposition, holds for (1.2) (Gordin and LifSic, 1978).

PROPOSITION 3.1. Assume p(x, dy) admits an invariant probability 11" and, under the initial distribution 11", (3.1) is ergodic. Assume also that (;=7j;-;j; is in the range of1 - T. Then

[

]

n-¹/2 ~(7j;(Xj) -;j;)+(nt - [nt])(7j;(X[ntl+l) -;j;) (t 2: 0) converges weakly to a Brownian motion with mean zero and variance parameter Ilhll~ -IIThll~, where (1 - T)h = {;, [nt] is the integer part of nt and;j;= J7j;d11".

Our main result is the following theorem.

THEOREM 3.1. Assume that the process in (3.1) is uniformly (geo- metrically) ergodic and that Xo has the unique invariant^11" as its distri- bution. Then for every7j; in L²(S,11") such that J7j;d11" = 0, 7j; belongs to the range of1 - T, i.e., (3.3) holds for every 7j; in L²(S,^11") such that

J7j;d11"= O.

Proof. Given the one-sided (strictly stationary) process {Xn : n ~

O}, we can always construct a two-sided {Xn : n = 0,±l, ±2,'" } with the same finite-dimensional distribution. We may call {Xn : n =

0, ±1, ±2,'" } the doubly infinite extension of {Xn : n ~ O} on a prob- ability space (n,F, P).

For a :::; b, define~ as the O"-field generated by the random variables X_{a , •..} ,X_b: define:F'!:-oo as the O"-field generated by ... ,Xa- 1,Xa : and defineF': as the O"-field generated by Xa ,Xa+1 , ' " .

We claim that for each m (-00 < m < 00) and for each n (n _~ 1), A E F".::oo, BE F::+ntogether imply

(3.4) IP(AnB) - P(A)P(B)I :::; cp(n)P(A), wherecp(n) is a nonnegative function of positive integers.

Let g(Xm+n)^denote E(IBIF.:::'~n),where IB is the indicator function ofB. Then

E[IAE(g(Xm+n)I~(x,)] ⁼ E[E(IAg(Xm+n)l~oo)]

E[IAg(Xm+n)]

= E[IAg(Xm)]

E[IAIB]

p(AnB).

On the other hand,

P(A)P(B) = E [lA / g(Y)7r(dY)] .

Therefore,

IP(AnB) - P(A)P(B)I

(3.5) = lE[IAE(g(Xm+n)l~oo)]- E [lA / g(Y)7r(dY)] I

=lE [lA ( / g(y)p(n) (x, dy) - / g(y)7r(dy»)]^I

:::; E[IAllglloollp(n)(x, dy) - 7r(dy)II].

Since the process{Xn :n ~O} is uniformly ergodic, there exist constants c >0, 0< p< 1, such that

1Ip(n)(x, dy) -1l"(dy)11 <cpn

for all sufficiently large n, uniformly for x E S. Thus, takingc:pn as our 'P(n), we have shown that for every sufficiently largen, (3.4) holds, and from the inequality of (3.5), a constant function <pen) can be taken for the other finite many number of n's, justifying our claim. Let 'ljJ ^E 11-, the set of all mean zeroL²-functions. Then

Ilrn'ljJll~==E( {E['ljJ(Xn)IXo]}2) (3.6) = E(E{E['ljJ(Xn)IXo]'ljJ(Xn)IXo})

=E(E('ljJ(Xn)IXo)'ljJ(Xn)) = E((rn'ljJ)(Xo)'ljJ(Xn))

~ p(n)II'ljJII~

(by definition of the maximal correlation coefficient p(n)).

Now consider the function g(y) := - L':=o (rn'ljJ)(y). Note that

00 00

IIgll2 ~ L IIrn'ljJ1I2 ~ 1I'ljJ1I2L J^{pen) < 00,}

n=O n=O

by the inequality (3.6). Hence9 E L²(S,1l"). But (T - 1)g = 'ljJ. Hence

'ljJ is in the range of(T - 1). 0

REMARK 1. The functional law of the iterated logarithm (FLIL) holds under the additional assumption J^1'ljJ12+<5d1l" < 00 for some 8> 0 (Bhat- tacharya (1982)).

REMARK 2. As references for earlier work on a class of nonlinear au- toregressive processes, see Bhattacharya and Lee (1988), and Nummelin (1984), where they have provided some verifiable conditions under which the functional central limit theorem (FeLT) holds for a class of functions

'ljJin L2_(lR¹_,1l").

ACKNOWLEDGEMENTS. The authors are indebted to the referee for many valuable suggestions.

References

[lJ Bhattacharya, R. N., On the functional central limit theorem and the law of iterated logarithm for Markov processes, Z. Wahr. Verw. Geb. 60 (1982), 185- 201.

[2] Bhattacharya, R. N. and Lee, C. H., Ergodicity of Nonlinear First Order Au- toregressive Models, J.Theoretical Probability 8 (1995a), No. 1, 207-219.

[3] , On geometric ergodicity of nonlinear autoregressive models, Statistics

& Probability Letters 22 (1995b), 311-315.

[4] Bhattacharya, R. N. and Lee, 0., Asymptotics of a class of Markov processes which are not in general irreducible, Ann. Prob. 16 (1988), 1333-1347., Correc- tion Note. Ann. Probab. 25 (1997), 1541-1543.

[5] Billingsley, P., Convergence of Probability Measures, Wiley, New York, 1968.

[6] Chan,K.S. andH.Tong, On the 'USe of the deterministic Lyapunov function for the ergodicity of stochastic difference equations,Adv. Appl. Probab. 17 (1985), 666-678.

[7] Gordin,M. I.and Lifsic,B.A., The central limit theorem for stationary ergodic Markov process,Dokl. Akad. Nauk. SSSR 19 (1978),392-393.

[8] Lee, C. H., Asymptotics of a class of pth-order nonlinear autoregressive processes, Stats. and Probab. Lett. 40 (1998), 171-177.

[9] Nummelin, K, General Irreducible Markov chains and Nonnegative Operators, Cambridge Univ. Press, New York, 1984.

[10] Tj<jlstheim, D., Nonlinear time series and Markov chains, Adv. Appl. Probab.

22 (1990), 587-611.

Chanho Lee

Department of Mathematics Hannam University

Taejon 306-791, Korea

E-mail: [email protected] Youngmee Kwon

Hansung University Seoul 136-792, Korea

E-mail: [email protected]