Large deviations for random walks with time stationary random distribution function

LARGE DEVIATIONS FOR RANDOM WALKS WITH TIME STATIONARY RANDOM DISTRIBUTION FUNCTION

DUG HUN HONG

1.

Introd uction

Let :F be a set of distributions on R with the topology of weak convergence, and let A be the <7-field generated by the open sets. We denote by :Fro the space consisting of all infinite sequence (F

F

F

E :F and R'f the space consisting of all infinite sequences (x

of real numbers. Take the <7-field A'f to be the smallest <7-field of sub- sets of :Fro containing all finite-dimensional rectangles and take Bf' to be the Borel <7-field of Rf'. Let

(Ft, F:;: , ... ) be the coor- dinate process in R'f and

its distribution on Ai. Let a be the coordinate shift :ak(w) = w' with F'::' = F:;'+k' k = 1,2,···. On (R'f, B'f) we also define the shift transformation <7 : R'f

R'f by <7(

=

is called stationary if for every

E A'f, lI(a- I

=

and we let

be its marginal distri- bution. Let I be the <7-field of invariant sets in A'f, that is, I

{Ala-I(A) = A, A E A l } ^{and let .J} be the <7-field of invariant sets in Bf\ that is, .J = {BI<7-

(B) = B,B

B'f}.

is called independent and identically distributed (LLd.) if

is stationary and product measure. For each w, define a probability measure P"" on (R'f,B l ) ^{so that} P"" =

monotone class argument shows that P",,(B),B

B'f, is Ai-measurable as a function

So we can define a new probability measure such that P(B)

J ^P

(B)II(dJ..v).

Define the process {X

on (Ri, B'f) such that Xn(XI'

and set Sn = Xl + X

+ ... + _{X n.} By the definition of P

{X n}

are independent with respect to P

and we also note that {X

is a sequence of independent and identically distributed random variables

Received February 23, 1994.

1991 AMS Subject Classifications: Primary 60FIO, 60G20; Secondary 60G50.

Key words and phrases: Large deviation, random walks, stationarity.

sequence of independent and identically distributed random variables when :F has just one. element. In this paper we generalize the classical Cramer theorem in this set up.

2. Strong law of large numbers

In this section we consider some strong law of large numbers which are used to prove the main results.

LEMMA

1. Let:F = ^{FI J ^{IxldF(x) < oo,J xdF(x)} = ^{O}, and let v}

be stationazy with JJ IxldF(x)7r(dF) <

Then Xl with respect to Psatisnes

E(X

1..71 ^{= 0} a.s.

P-roof. By the assumption, EfX

I <

and henceE(X-rlJ} exists.

Now let A

..7

let {(XI,X2,"')

B} = ^{A for} some B

Bf'.

Then we have

wher~.th~

last

,llolqs beca¥l;e J ^{xdF(x) = 0 for all F E}

^This

proves the lemma.

THEOREM

1. Let:F

{FI J ^{xdF(x) =} O,J ^IxldF(x) <oo} and v be stationary with J J IxldF(x)7r(dF) <

Then

p",{~ -+o} ^{= 1, v-a.e. w.}

Proof. TJie proof follows directly from Proposition laIid3[5],

Lemma

1, and the Birkhoff's ergodic theorem.

In general

then prove the following theorem.

v - a.e.

THEOREM

2. Let:F = {FI J IxldF(x) < oo} and let v be stationary with JJ IxldF(x)1l"(dF) <

Then

Pw{ -;

E[f xdFi(x)IZ] (w)} = 1,

(E[J xdFi(x)II](w)

E[J xdFf(x)]

JJ xdF(x)1l"(dF) in case v is ergodic.)

Proof. By Theorem 1, Pw{ Sn-~!t/Sn

^D}

^{1, v-a.e.}

^where EwS

= 2::=1 f XkdPw

E:=1 J xdFk'(x). We know ~EwSn

E[J xdFi(x)II](w),v-a.e.

by the ergodic theorem. Hence v - a.e.w.

3. Large deviations

We begin this section by introducing the logarithmic moment gener- ating function

=

where

= J

E

R, and C(e)

J1" CF(()7T(dF), (

R. Throughout this section we

assume

(3.1) Mp(() <

for allF

F and for all(

R,

(3.2)

C(O <

for alle

Note that since e

^R

is a convex function, for each F

F, so is C(e). Next let K(x) be the Legendre transform of C(O :

(3.3) K(x) == sup{ex - C(e)/e ER},

ER.

Note that, by its definition as the pointwise supremum of linear func-

tions, K ( x) is necessarily a convex function. In order to develop some

feeling for the relationship between C(O and K(x), we present the

following elementary lemma.

v-a.e.w.

LEMMA

2. K( x)

0, moreover, if J J ^IxldF(

^{)?r( dF) <}

^{and p}

JJ ^xdF(

^)?r( dF) then K(p)

0, K is non-decreasing on fp, (0) and non-increasing on (-oo,p]. In addition, for q

p, K(q)

sup{eq - G(e)le

O} and for q :5 p, K(q)

sup{eq - G(e)le :5 O}.

Proof. We begin by noting that, since ex - G(e) = 0 for e = 0 and for every x E R, Kee)

O. Now suppose that JJ IxldF(x)?r(dF) <

and set p =J J xdF(x)?r(dF). To see that K(p) = 0, we use Jensen's inequality to obtain

G(e) = L(log f eXPCex)dF(x»?r(dFJ

~ L ^{f exdF(x )?r( dF) = ep for all e}

^R.

In particular, this shows that ep - C(e) s ^{0 for an e}

^{R and hence}

K(P) :5 O. Since K(x) is non-negative and convex, this proves that K(p) = ^{0, that} K(x) is non-decreasing on fp,oo), and that K(x) is non-increasing on (-oo,p].

As a consequence of Lemma 2, we have the following.

- LEMMA

3. Let:F = {FI f exp(ex)dF(x) < Oo,e ER}. If v is stationary and ergQdic with J J IxldF(x)?r(dF) <

then for every' closed set G c R,

limsup .!.logpw{ Sn

G} :5 - mf{K(x)lx

G},

n-oo n n

Proof· Let p = J J xdF(x)1r(dF). Suppose q

p(q:5 p). For e

0, Pw{'; ~ ^q}:5 exp(-eq)Ewexp(e~n)

= exp( -eq)I1i=l E

exp ( e~i) .

Note that

v-a.e.

by the ergodic theorem. Then for given e > 0, and v-a.e.

that we have for n

N(w)

; logpw{ ^:n ~ ^q} :s inf{ -eq + C(e)le ~ ^{O} + e.}

Since e is arbitrary, by Lemma 2,

limsup.!.logPw{Sn "? q} :s inf{-eq + C(q)le"? O} = -K(q).

n-oo n n

Since K is non-decreasing(non-increasing) on lP, 00)(on (-oo,p]) above inequality proves the result when either G c lP, 00) or G c

(-oo,p]. On the other hand, if both G n [p, (0) =F fjJ and G n (-oo,p] =F fjJ, let q+ = ^{inf{x ;::: plx}

^{G} and} q_ = sup{x $ plx

G}. Then for

el ~ 0, e2 "? °

pw{:n

G}

:5 exp( -6q+)E

(ex

^{(6 -;)) + exp(6q_)E}

(exp(-6 ^:n))

and hence

lim sup .!.log Pw {Sn

G}

n-oo n n

:5 max [lim sup.!. log (eXP( -6 q+ )E

exp( 6 Sn ») ,

n-(Xl

n n

lim sup.!. log (exp( _n-oo

)E

exp( -6 Sn »)]

n n

:5 max[-K(q+),-K(q_)]

-inf{K(x)lx

G}.

For the lower bound we need the following lemma. We define F-1(t)

sup{xIF(x) :5 t}, t E (0,1).

(3.4)

LEMMA 4. Suppose tbat for every F E

F-

is .unbounded below and above and tbat tbere exists a measurable function 4J( e, F) such tbat

1/ x~X:i~) ^dF(x)1 ~ ^{4J(e, F),}

(3.5) / (sup 4J(e,F») 7r(dF) < 00 for an eo ER.

1(1~(o

Tben webave

i) j(e) = J/ ^x ~X:i~;) ^dF(x )7r(dF) is continuous and

f( e)

= ±oo,

ii) For each q, K( q) = ^8Up{ eq - c( e)le E R} is assumed at some point' e =:= ec(q) or equivalently C'(ec(q» = q.

Proof. i) We know that for all F, the function e ~ ^J x::i~;) ^dF(x)

is contihuous and J x ::~~;) ^{dF( x)} ~ ^{+00(-00) as} e _~ ^{+oo( -'-(0) by}

unboundedness of F-

So f(

is continuous by the Lebesgue dom- inated convergence theorem using (3.4) and (3.5) and we can easily check fee)

+00(-00) as e

+00(-00).

ii) consider the following:

lim G(e) - G(e')

(--(' e- e'

= fun J GF(e)-CF(e')7r(dF)

(--(' e...:. e'

= i!!:i, J GF«()7r(4F), where (E (e,e') or (E (e', e)

= lim JJ ^xexp«( x) dF(x)7r(dF) e--(' MF(e")

f · ^{J xexp«( x)}

= .... i~, MF(e',)dF(x)1r(dF)

= JJ x::i::·;) dF(x)7r(dF) = f(e');

hence

=

The fourth equality above follows from (3.4), (3.5) and the domi- nated convergence theorem. Note that G is convex. So for every given q there exists

such that

= q. This proves the lemma.

THEOREM

3. Suppose that v is stationary and ergodic. Then under (3.1), (3.2), (3.4) and (3.5) for every measurable re R we have that

- inf{K(x)lx

rO}

< liminf.!.logPw(Sn Er) ~ limsup.!.logPw(Sn Er)

n-oo

n n

n n

< - inf{K(x)lx

f} v - a.e.

Proof.

In view of Lemma 3, we only need to show that if

E Rand 6> 0,

(3.6) liminf .!.log Pw (Sn

(q - E, q + E») _~ -K( q).

n-oo n n

In proving (3.6), we first suppose that for all F E:F, F-

is unbounded above and below. Then for each q, there exists e such that G'(e) = fee) = q by Lemma 4, and so K(q) = eq - G(e).

pw{:n E(q-6,q+6)}

= J f{ J

±.~±"'"'

} dFl'(Xl)' .. dF:(xn)

~

MFf(e)· ..

exp( -e(q + 6)n) x !{ ^.,+,;+•• E(,-,.'+6)}

exp(ex,) dF

exp(ex

dF

MFf(e)

MF;:-(e)

Here we need the following lemma.

w

LEMMA

5. Under the conditions of Theorem 3, we have for v-a.e.

as n

. . jt ^exp(ex)

Proof· For given e define F so that F(t) =

MF(e) dF(x). Let j:

{FIF

.r}. Define 4>:.rro

Fro by 4>(w) =w ⁼ (Ff,F7:,"')'

Now let v ⁼ v 04>-1. Then v is stationary and ergodic. Now we apply Theorem 2 to this probability measure. Then we have

P';'{';

JJ xdF(x)1r(dF)}

1 v - a.e. w, Note that JJ xdF(x)1r(dF) = JJ Z;;F~t; dF(x)1r(dF) = ^{f(e) = q.}

Hence the lemma follows.

Now back to the proof of Theorem 3. By the above lemma we have, for given

> 0, and v-a.e. w that for

N(w)

1 S I n

;; logPc.1{ nn

(q - 8,q + 8)} ~ ^;; L)ogMFt(e) - e(q + 8) -

;=1

and consequently

liminf.!.logpc.1{Sn E (q-8,q+8)} ~ C(e)-e(q+8),

n-+oo

n n v-a.e.w.

By monotonicity, the result holds with 8 = 0, i.e., with K(q).

We must now handle the general case. Suppose that there exists FE .r such that F-

is bounded. We replace all F by the distribution

1 {Z!€ ^y2

F *4>£ where4>€(x) = - exp{-- )dyand apply the above

$

2

results to this. Here * is the convolution. Then, letting

1 ° ^the

desired result follows.

REMARK

1. If

is LLd., then {X

is i.i.d. with respect to P with distribution function FI(x) = J F(x}7r(dF). By the Cramer theorem, {X

with respect to P has the rate function

K(x) = sup{ec - C(e)le ER}.

where G(e) = ^log J exp(ex)dF(x) = ^log JJ exp(ex)dF(x)7r(dF). By Jensen's inequality, we can check easily C(e)

C(e) and hence K(x)

K(x).

References

1. L. Brieman, Probability, Addison-Wesely, Reading, Mass, 1968.

2. K.L. Chung, A course in probability th.eory, 2nd ed., Acadimic Press, New York London, 1974.

3. J.D. Dueschel and D. W. Strook, Large deviations, Academic Press, 1989.

4. D. H. Hong, Random .Gll:s with time stationa", nJndom distribution junction, Ph.D. Thesis, Univ. of Minnesota., 1990.

5. D. H. Hong and J. S. Kwon, An LIL for random walks with. time stationary nJndom distribution junction, Yokohama Math. J. 40 (1993), 115-120.

Department of Statistics

Taegu Hyosung Catholic University

Kyungbuk 713-702, Korea