Limit theorems for some doubly stochastic processes

STATISTICS& PROBABILITY LETTERS

ELSEVIER

Statistics & Probability Letters 32 (1997) 215-221

Limit theorems for some doubly stochastic processes Oesook Lee Department of Statistics, Ewha Womans University, South Korea Received December 1995; revised April 1996

Abstract We consider the stochastic processes Xk+l = Fk+l(Xk)+ Wk+l where {Fk} is a sequence of nonlinear random functions and {Wk} is a sequence of disturbances. Sufficient conditions for the existence of a unique invariant probability are obtained. Functional central limit theorem is proved for every Lipschitzian function on R.

Keywords." Doubly stochastic process; Markov process; Invariant probability; Weak convergence; Functional central limit theorem

In this paper we consider the limit theorems for some discrete time doubly stochastic systems {Xk} on R which is given by Xk+l = Fk+l(Xk)+ Wk+l (k >~O).

(1)

We make the assumptions: (A1) {Fk: k ~>1} is a sequence of independent and identically distributed random elements taking values on F with common distribution Q, where F is a collection of Borel measurable functions on R, each of which is bounded on compact sets. For any x E R, EIFI(X) [ < c~. (A2) { Wk: k/> 1} is a sequence of independent and identically distributed random variables with E[ Wt I < c~. (A3) Initial X0, {Fk} and {Wk} are mutually independent. Then {Xk: k >/0} generated under the above assumptions is Markovian with one-step transition probability function p(x,A) = Pr(Fk+~(Xk) + Wk+l ~ AIXk = x),

(x E R,A E ~ )

where M is the Borel sigma field of R. Let 2 be a Lebesgue measure on R.

I This research is supported by Ewha Womans University Faculty Research Fund, 1994. 0167-7152/97/$17.00 (~) 1997 Elsevier Science B.V. All fights reserved PH S 0 1 6 7 - 7 1 5 2 ( 9 6 )00076-4

O. Lee / Statistics & Probability Letters 32 (1997) 215-221

216

The Markov process {Xk: k~>0} is 2-irreducible if

Z p"(x,A) > 0 n>~l

for all x E R, whenever 2(A) > 0. Here pn(x, dy) denotes the n-step transition probability. A 2-irreducible Markov chain with transition probability p(x, d y ) is said to be 9eometrically Harris ergodic if there exist a probability measure rc and a constant p, 0 < p < 1, such that p-nil pn(x,.) - ~(.) II ~ 0

as n ~ ~ ,

V x ~ R.

(2)

Here ][. [[ denotes the total variation norm. A probability measure r~ on ( R , ~ ) is said to be invariant for {Xk: k~>0}, or p(x, dy), if it satisfies

f p(x,A)g(dx) = 7r(A), VA E 8. If (2) holds, then 7t is necessarily the unique invariant probability for p(x, dy). For unexplained notations, we refer the reader to Nummelin (1984). When time series has an irreducible Markovian structure, following result due to Tweedie (1983) gives sufficient conditions for geometric ergodicity. Theorem 1. Suppose {Xk : k>~0} is a 2-irreducible Markov process. If there exist a measurable function V >~1 on R and positive constants M1, ME and 6 such that (i)

sup

[ V(y)p(x, dy) < oo

(3)

xE[-M2,Mt] J

(ii)

/ V(y)p(x, dy) < (1 - 6)V(x),

Vx E [-M2,M1] c,

(4)

then the process is geometrically Harris ergodic. Let F+(x) = max {F(x),0}, F-(x) = max { - F ( x ) , 0 } , and define ~1 =

lim sup

Er+(x)

x-.-~

Ixl

,

EF+(x)

//1 = lim s u p - - , X--*CX~

c~2 =

lira sup

Er-(x)

x-.-~

Ixl

EF-(x)

//2-- lira s u p -

X

X----~

X

(5)

(6)

Theorem 2. Let the assumptions ( A 1 ) - ( A 3 ) hold. In addition, assume the distribution F of Wk has an

absolutely continuous component with an almost everywhere positive density q(.) with respect to 2. Then the process {Xk: k i> 0} is 9eometrically Harris ergodic if (i) (ii)

fll < 1, (1

-- 0(2)(1 -- ill)

(7) >

~1//2.

Proof. Since

p(x,A)>~ fr ~ q(y - 7(x))dydQ(7) > 0 for every x C R and A E ~' with 2(A) > 0, {Xk } is 2-irreducible.

(8)

O. Lee I Statistics & Probability Letters 32 (1997) 215-221

We define the Lyapunov function 1 +plx

V(x) =

217

V(x) by

ifx~>0, if x < 0,

1 - pzx,

where pl and P2 are positive constants which will be chosen later. Now

/RV(y)p(x, d y ) = / ; f R V ( Y + 7 ( x ) ) d F ( y ) d Q ( Y ) (l + p , ( y + y(x))dF(y)

=fr[f{

y+7(x)~>O}

+ f{y+7(x)<0}(1 -- P 2 ( Y + 7(x))dF(y)IdQ(Y) >i O}

dF(y)dQ(7)

-P2frT(X)fy+,(x,10, ti>10, i = 1,2. We first choose and fix 6 > 0 sufficiently small such that

fll+b

< 1,

~2-[-b < 1

and 1

-

-

~2

>

el

1 - (~2

+ 6)

e1+6

>

1-

fl2+6 (i1+6)

>

f12 1 - i1"

-

Then choose pl, p2 C (0, 1) such that 1 - (~2 + 3) > -Pl - >

~1 + 6

P2

i2 + 6

(9)

1 -- (ill + 3)"

There exists M0 > 0 so that Vx > M0,

V(y)p(x, dy) <<.c + p~E(r+(x)) + p2E(r-(x)) ~
= c + Ooplx, where Oo = (ill + 6 + (p2/Pl)(fl2 + 6)) < 1, by (9). Next choose M1 > Mo and 00 < 01 < 1 such that Vx

>

M1,

V(y)p(x, dy)<.Ol(1 + plx).

(10)

Similarly, we may choose M2 > 0 and 02 < 1 such that gx < - M 2 ,

f V(y)p(x, dy)<~02(1 + P2lx[).

(11)

O. Lee / Statistics & Probability Letters 32 (1997) 215-221

218

If we set 0 = max{01,02}, then 0 < 0 < 1, and from (10) and (11),

Z(y)p(x, dy)<.OV(x),

Vx E [-M2,M1] c.

From the condition that each 7 E F is bounded on compacts, sup

[ V(y)p(x, dy) < oc.

xE[--Mz,MI ] J

Thus, (3) and (4) in Theorem 1 hold, and hence {Ark} is geometrically Harris ergodic. In Theorem 2, we assume the ).-irreducibility of {Xk } but sometimes, to check the irreducibility is awkward or even given process is not irreducibile. For example, {Xk} given by Xk+l = Fk+l(Xk) is not irreducible when the distribution of {Fk} has a finite or discrete support. There are some papers in which the sufficient conditions for the existence of the unique invariant probability are given without irreducibility condition. (see Bhattacharya and Lee, 1988a; Tweedie, t988). Let B(R) be the set of all bounded real-valued uniformly continuous functions on R. Note that if every 7 E F is continuous, then {Xk} becomes a weak Feller process. For following theorems we let {~k} be a sequence of independent and identically distributed random variables, each having a standard Normal distribution N(0, 1 ) and let {~k}, {Fk}, and{Wk} are mutually independent. We shall write Xk(x) for Ark in case X0 = x. We prove the following theorem which is for nonirreducible case by considering perturbed system first, and then using the approximation technique.(Meyn, 1989). Theorem 3. Under the assumptions (A1)-(A3), the process {Xk} has the unique invariant probability measure if there exists a measurable function M on F such that (i)

V7 ~ F, IV(x) - ~'(Y)I ~
(12)

(ii)

E(M(Fl)) < 1.

(13)

Proof. We define a perturbed process {X~ : k~>0} for each e E [0, 1] which is given by X~ =Xo, and X~+ 1 = Fk+l(X;)-q- Wk+x q- E (~k+l

(k~0).

Then for each ~ > 0, {X~} is 2-irreducible and aperiodic. Since, q = E(M(F1)) < l, the conditions (7) and (8) in Theorem 2 are satisfied. Hence by Theorem 2, for each e E [0, 1], {X~} is geometrically Harris ergodic, and let u~ be as its unique invariant probability such that lim E [h(X~(x))] = [ h d u ~, J

k~oo

for h E B(R).

(14)

Now from(12) and (13), we get

EIX~(x)I

~,?lxl +

(1 - q)-lK,

EIX~(x ) - Xk(x)l ~
(15) (16)

O. Lee / Statistics & Probability Letters 32 (1997.) 215-221

219

From (14) and (15), {1if: e E [0, 1]} becomes a tight set and therefore there exist a subsequence en ---+0 as n ---+oo and a probability measure ~ on D~ such that ~" --+ it weakly

as n ---+oo.

(17)

Using (16) and Chebyshev's inequality, we have for h E B(R), lim sup E[lh(Xk~(x)) -- h(Xk(x))]] = 0.

(18)

c---+0 k>~0

But for each positive integer n, li~n sup E [h(Xk(x))] - S h drc ~< lim sup

IE [h(Xk(x ) )]

- E [h(Xk~"(x ) )] I

k--+ex)

+lim___+supE[h(Xk~"(x))]-/harts"+ i hd.~"-Shaft . Combining (14), (17), (18) and letting n --+ oo shows that lim E [h(Xk(x)] = f l h drt. J

(19)

k---+o,o

Suppose there is another weak limit point re' of {rc~: ~ E [0, 1]}, then by (19), fhdTt = fhdrc' for every h E B(R), and hence rc = g'. Since {Ark} is a Feller process, the unique weak limit point rc is the invariant probability measure for {Xk}. Remark. Theorem 3 improves Theorem 2.2 in Bhattacharya and Lee (1988b) specialized to the case of the state space R. The result in that article assumes Wk = 0, and contraction for all maps in the support of Q. In order to state the functional central limit theorem (FCLT), fix f E L2(R,n). Define Tnf(x) = f f

(y)pn(x, dy), T f ( x ) = Tlf(x). For each positive integer n, write L~=0 F[nt] yx(t)=n-'121S-'(f(Ss(x))f ) + ( [t n- t ] )

(f(Xf.,,+l(x))-f)

, (t>~O),

n

where f = f f d r t and [nt] is the integer part of nt. Theorem 4. Let the hypotheses of Theorem 3 hold and let f x 2 dg < oo for invariant measure ~. Then

whatever the initial x, for every Lipschitzian f , the process yx(.) converges in distribution to a Brownian Motion with mean zero and variance parameter Ilgll 2 - II~ 9112, 2 where 7"9 - g = f - f . Proof. Let f be a Lipschitzian on R, If(x) - f(Y)l <~K]x - y] for all x, y. It is proved (see Theorem 2.5 in Bhattacharya and Lee, 1988b) that if ~-~ IITk(f - f ) l l 2 < oo, k=0

then the conclusion follows. But

[]Tk(f - f )ll2 <~K2 i

[f E lXk(x) - Xk(y)l dg(Y)] 2 dg(x).

O. Lee / Statistics & Probability Letters 32 (1997) 215 221

220

Moreover from (12) and (13), we have E IXk(x) - Sk(y)l ~
~-~llTk(f -/)ll2 <<.Ky-~ nk k=0

J ( x - y)2 d~(y) d~(x)

< 0(3.

(20)

k=0

The second inequality of (20) follows from the relation 1

= f

x2dg(x)

-

Note that every Lipschitzian function is in L2(R, n) if f x 2 dn(x) < oc. Corollary 1. Under assumptions of Theorem 3, if we assume E IW112+6 < oe and E[M(Y1)] 2+~ < l for some 6 > O, then for any initial x, F C L T holds for every Lipschitzian f on R.

Proof. Using Minkovski's inequality, we can easily have

supE [lXk(x)[ 2+6] < oo.

(21)

k>~O (21) and weak convergence of pk(x,.) to ~(.) implies f x2drc(x) < cx2. Hence by Theorem 4, the conclusion follows. Write F m for the set of all compositions 7172-..Tm of elements 7i E F, (i = 1,2 . . . . . m). Corollary 2. We consider the model {Ark} defined by Xk+t = Fk+l(Xk) where {Fk} satisfies the condition (A1). Assume each 7 E F is continuous and initial Xo is independent on {Fk}. I f there exist an positive

integer no and a measurable function M on F n° such that (i) (ii)

Vy E F "°, [?,(x) - 7(Y)[ EM(F n°) < 1,

<~M(v)lx-

Y[,

then there exists a unique invariant probability rcfor {Xk} and F C L T holds for every Lipschitzian function if f x2 drc < oc. Proof. Let Y0 = X0, and Yk+l = F~°+l(Yk) where F kno = F(k_l)no+l ...Fkno. Then the function q(x, dy) = pn°(x, dy) becomes a one-step transition probability function for {Yk}. By Theorem 3, there exists a unique invariant probability n for q(x, dy) and n is also a unique weak limit of qk(x, dy), which implies the weak convergence of pk(x, dy) to zc as k ~ e~, and therefore the Feller property of {Xk} shows that n is the unique invariant probability for p(x, dy) . Moreover, we have

E IXk(x) -

Sk(y)l ~ ~l[k/n°]ElXi(x) - ~ ( y ) [ ,

(22)

O. Lee I Statistics & Probability Letters 32 (1997) 215-221

where

i=k-[k/no],

221

and

f E(Xi(x))2dn(x) =

f

X2

dn(x).

(23)

From (22) and (23), we obtain (20), which completes the proof. References Bhattacharya, R.N. and Lee, C. (1995), Ergodicity of nonlinear first order autoregressive models, J. Theoret. Probab., 8, 207-219. Bhattacharya, R.N. and O. Lee (1988a), Asymptotics of a class of Markov processes which are not in general irreducible, Ann. Probab. 16, 1333-1347. Bhattacharya, R.N. and O. Lee (1988b), Ergodicity and central limit theorems for a class of Markov processes, J. Multivariate Anal 27, 80-90. Bhattacharya, R.N. and E.C. Waymire (1990), Stochastic Processes with Applications (Wiley, New York). Billingsley, P. (1968), Convergence o f Probability Measures (Wiley, New York). Feigen, D.F. and R.L. Tweedie (1985), Random coefficient autoregressive processes: a markov chain analysis of stationarity and finiteness of monents, J. Time Series Anal. 6, 1-14. Meyn, S.P. (1989), Ergodic theorem for discrete time stochastic systems using a stochastic Lyapunov function, S l A M J. Control Optim. 27, 1409-1439. Nummelin, E. (1984), General Irreducible Markov Chains and Nonnegative operators (Cambridge Univ. Press, Cambridge). Tjostheim, D. (1986), Some doubly stochastic time series model, J. Time Series Anal. 7, 51-72. Tweedie, R.L. (1983), Criteria for rates of convergence of Markov chains with applications to queueing theory, in: J.F.C. Kingman and G.E.H. Reuter eds., Papers in probability, Statistics and nonnegative operators (Cambridge Univ. Press, Cambridge). Tweedie, R.L. (1988), lnvariant measures for Markov Chains with no irreducibility assumptions, J. Appl. Probab. 25A (A celebration of Applied probability) 275 285.