ELSEVIER Statistics & Probability Letters 21 (1994) 157-161

STATISTICS&

A strong limit theorem under no assumption of independence, stationarity or various dependences

Wen Liu

Received April 1993; revised June 1993

Abstract

Let (q,,, n 2 0) be a sequence of positive integers, I, = {0, 1, . . . . qn}, {Xnr n > 0} a sequence of random variables, where

X, takes on values in I,, and P(X, = x1, . . . . X, = x,) > 0, for all xi E Ii, 0 < i < n. The purpose of this paper is to give a strong limit theorem for the above sequence of random variables concerning conditional expectation without any assumption of independence, stationarity or various dependences.

Keywords: Strong limit theorem; Strong law of large numbers; Conditional expectation

In previous works, strong limit theorems were discussed under the assumptions of independence, station- arity or various dependences. The purpose of this paper is to avoid these assumptions and give a strong limit theorem concerning conditional expectation.

Theorem. Let {q,,, n 3 0) be a sequence of positive integers, I, = (0, 1, . . ..q.,), {X”,n 3 0) be a sequence of random variables, where X,, takes on values in I,, and

P(XIJ = X0, . . . , X, = X,) = p(X0, . . . , X,) > 0, for all Xi E Ii, 0 d i < n. (1)

If

jI qn2/n2 < a y (2)

then

lim(l/n) i [Xi- E(Xi(X(),...,Xi-I)] =O a.e. n i=l

(3)

Proof. Throughout this paper we shall deal with the underlying probability space ([0, l), 9, P), where 9 is the class of Bore1 measurable sets of the interval [0, l), and P is the Lebesgue measure. We first give, in the above probability space, a realization of the sequence of random variables with distribution (1).

0167-7152/94/$7.00 0 1994 Elsevier Science B.V. All rights reserved SSDI 0167-7152(94)00021-Y

158 W. Liul Statisiics & Probability Letters 21 (1994) 157~161

Divide the interval [0, 1) into q. + 1 right-semiopen intervals:

2i0 = [0,&O)), G,(and so on).

These intervals will be called Oth-order intervals. Proceeding inductively, suppose the (q,, + 1) ... (q. + 1) intervals {6x, ... x,, xi = 0, 1, . . . . qi, 0 ,< i < n) of order n have been defined. By dividing the right-semiopen interval 6x0 ... x, into qn + 1 + 1 right-semiopen intervals 6x0 ... x,x, + r (x, + i = 0, 1, . . . , qn + 1) at the ratio

P(xo, . . . > &nO):ph . . . . .%,1): . . ..P(Xo....,Xnr4n+l),

the intervals of order (n + 1) are created. For n 3 0, define a random variable X,,: [0, 1) + S as follows:

Xn(w)=xn, ifwEi3x0...x,. (4)

It is easy to see that

P(X, = xg, . ..) X” = x,) = p(Gxo . . . x,) = p(xo, . . . ) x,) (5)

and (X,, n 3 0} has distribution (1). Let the collection of intervals of all orders and the interval [0, 1) be denoted by d, and g be a Bore1

measurable function. Denote

P(x,Ixo,xr, . ..> X,-i) = P(X, = x,1x0 = xg,...,xn_i = x,-i);

&J(XJlxo> ...3 X,-i) = E(g(X,)lX, = x0, . ..) x,-i = x,-r).

Assume 1, = 1 or - 1. Let

d,(xo, . . . , x,-~) = E{expCW, - WGlxo, ...,~,-l~~/~lI~o,...,~,~l}

= $, P(xnlxo> . . . . x,-I)exp{ACx, - ~(X,lx~,...,x~-~)ll~}. n

Define a set function p on 1;4 as follows: if n 3 1, let

/@x0 ... x,) = P@xo ..~Xn)~~=,eXp(~[Xi-E(XilXo,...,Xi-l)]/i}

n~=,~i(XO,*..,Xi-l)

And let

,Gxo) = F /@x0x1); XL =o

.dco> 1)) = f PL(~Xo). .X0=1

As n > 1, we have by (7),

p(6xo ... x,) = p(Gxo . . . Xn-1)P(XnlXO, . ..t x,-l)exp{ACx, - EGGlx0, . . ..x.-1)1/n)

cJn(xo, . . ..&I)

(6)

(7)

(8)

(9)

(10)

BY (6) and (1%

.$, p(Fxo ... x,) = p(6xo ... x,_ 1), n 2 1. n

(11)

W. LiulSiatistics & Probability Letters 21 11994) 157-161 159

By (8) (9) and (1 l), p is an additive set function on d. Hence there exists an increasing functionfn defined on [0, 1) such that for any 6x0 ... x,,

/@xo...x,) =f~(6+xo,..x,)--f~(~-xg...xn), (12)

where 8+x0 ... x, and 6-x0 ... x, denote, respectively, the left and right endpoints of 6x0 ... x,. Let

40,~) = P@Xo ... --%I) J2s +x0 ..’ x,) -fA(s-x, ... x,)

P(Fxo ... x,) ti+x~...x,-~-x()~~~x, ’ Co E 6x0 ... x,. (13)

Let A(1) be a set of points of differentiability off,. Then (cf. Billingsley, 1986, p. 423)

lim t,(i,co) = a finite number, o E A(A), n

(14)

and P(A(i)) = 1 by the existence theorem of derivative of the monotone function. By (13) (7) and (4)

t (n w) = nf=, exp{2[Xi - E(XilXo, . . ..Xi-l)I/iI n 7 JJ~=rGi(X,,...,Xi-r) ’ wECo’l).

(15)

Noting that

,tO [x, - E(X,IXC,, . . ..Xn-.)l~h,l&, . . ..X-I) = 0, n

we have by (6) and the inequality 0 d ex - 1 - x d x2elxt,

0 d a,(X(),...,X,_r) - 1

= .t, {expC(Vn)(x, - E(X,IX,, . . ..LI))] - 1 - W)Cx, - WLI&, . . ..x.-~)l) n

x Pc%l&, . . ..&I)

4n d c (lln”)Cx, - E(KI&, . ..> LJ12expCU141xn - E(XI&,...,Xn-~)Il

x,=0

xP(xnI&, . ..>LJ

d (aJ~)2~v~~n14.

We have by (2) q,/n + 0, and it follows from (2) and (16) that

jj a,(Xo> . . . , X, 1) converges.

We have by (14), (15) and (17),

lim fi exp{A[Xi - E(XiIXo, . ..>Xi-l)]/i} = hm exp n i=l n

i$I(A/i)[Xi - E(XiIX~,...,xi-l)l)

= finite number, o E A(%).

Letting A = A(1) n A( - l), we have by (18)

$I (l/i) [Xi - E(X, 1x0, . . ..Xi-t)] converges, oEA.

(16)

(17)

(18)

(19)

160 W. Liul Statistics & Probability Letters 21 (1994) 157p161

By (19) and Kronecker lemma,

lim (l/n) i [Xi - E(XJX,, . . ..X._i)] = 0, 0 E A. II i=l

(20)

Since P(A) = 1, (3) follows from (20). This completes the proof of the theorem. 0

Letq,>2((n= 1,2,... ). It is well known that every real number w E (0,l) can be represented in the form

w= f X(4

n=l q1q2 ...qn (21)

where the nth digit X,,(w) can take on the values 0, 1,2, . . . , qn - 1. The series (21) is called Cantor series. (The representation is unique, except for some rational numbers.)

Let X0 = 0, o E (0,l). It is evident that

E(X,lX,, . ..> X,-i) = E(X,), n = 1,2 ,... . (22)

Hence we have the following corollary.

Corollary. Let qn 3 2 (n = 1,2, . ..). and X,,(o) be the nth digit ofw E (0,l) in its Cantor’s series (21). Zf

,,tl q:/ n2 < cc, a.e. (23)

then

lim (l/n) i [Xi - E(XJ] = 0 a.e. ” i=l

(24)

This is the strong law of large numbers for (X,, n > l}. If qn is a constant, the above corollary follows from a well-known result on normal numbers (cf. Feller, 1957, pp. 195-197). For the general case of q. the statistical properties of the sequence Xi(w), X,(o), . . . in the Cantor series (21) were investigated by Renyi

(1956) (cf. Reverz, 1968, p. 52).

Remark 1. In the above corollary {X,,, n > l} are independent random variables possessing discrete uniform distribution with Var(X,,) = (qi - 1)/12, and it is easy to see that the condition (23) is equivalent to that of Kolmogorov’s strong law of large numbers (see Halmos, 1974, p. 204).

f Var(X,)/n2 < co. n=l

(25)

It is well-known that for the strong law of large numbers of the sequences of arbitrary independent random variables the condition (25) involving restrictions on variances cannot be weakened in some sense. (cf. Halmos, p. 204, Problem (3))

Remark 2. The following weak converse of the above corollary is true. If there exists a positive constant c such that q,/n d c, n = 1,2, . . . . and if (24) holds, then

nF1 q,2/n2+’ < a (26)

for every positive E. (cf. Halmos, 1974, p. 205, Problem (5))

W. LiulStatistics & Probability Letters 21 (1994) 157-161

Acknowledgement

161

The author is thankful to the referee for his helpful suggestions.

References

Liu Wen (1990), Relative entropy densities and a class of limit theorems of the sequence of m-valued random variables, Ann. Probab. 18 (12), 829-839.

Billingsley, P. (1986), Probability and Measure (Wiley, New York).

Feller, W. (1957), An Introduction to Probability Theory and Its Applications (Vol. 1, 2nd edn., Wiley, New York).

Rtnyi, A. (1956), Distribution of numerals in the Cantor productions of real numbers, Mat. Lapok 7, 77-100. Revesz, P. (1968), The Laws of Large Numbers (Academic Press, New York).

Halmos, P.R. (1974), Measure Theory (Springer, New York).