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Letters

27 January

16 (1993) 163-168

1993

A note on moment bounds for strong mixing sequences Tae Yoon Kim Department of Statistics, Keimyung University, Taegu, South Korea Received August 1991 Revised April 1992

Abstract: For a g > 2 and sequence {X,) under less can be applied directly to limiting laws for a-mixing

finite K, E IE~=,X, I g -c , fig/2 (all n > 1) is obtained for a strictly stationary strong mixing or a-mixing restrictive conditions than Yokoyama’s (1980). Doob’s argument (1953) for +mixing sequences of r.v.‘s a-mixing sequences and leads to improved results for moment bounds when g > 4. As applications, some are discussed.

AMS 1980 Subject Classifications: Primary Keywords: Strong

mixing sequence;

60E15; Secondary

stationarity;

moment

60F15.

bounds.

1. Introduction Moment

bounds

g

for various

types of random

II

>

variables

have been obtained.

For g > 2 and finite

K,

1,

(I.11

has been studied if either {Xi) is (i) a sequence of mutually independent r.v.‘s; or (ii) a stationary Markow sequence satisfying Doeblin’s condition; or (iii) a strictly stationary &mixing sequence; or (iv) a strictly stationary a-mixing sequence. Detailed discussion may be found, for example, in Brillinger (19621, von Bahr (1965), Doob (19531, Yokoyama (1980). This type of bound has proven to be of considerable use in obtaining several types of limit laws, notably central limit theorems and strong laws: see, for example, Lemma 7.4, p. 225, in Doob (1953) and Theorem 3.7.7, p. 211, in Stout (1974). However, Yokoyama’s (1980) results for a-mixing require more restrictive conditions than results for other mixing r.v.‘s, mainly because his proof could not extend in a straightforward manner Ibragimov’s (1962) proof for +-mixing to a-mixing, due to the difference between the basic moment inequalities (2.1) and (2.2) below. Inequality (1.1) holds for a stationary +-mixing, with EX, = 0, E) Xi I g

to: Tae Yoon

0167.7152/93/$06.00

Kim, Department

0 1993 - Elsevier

Science

of Statistics,

Publishers

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of Natural

B.V. All rights reserved

Sciences,

Keimyung

University,

Taegu,

South

163

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is sufficient to impose the condition on mixing coefficients so that E(Cy=,Xi)2 ( Kn, which is not the case for a-mixing according to Yokoyama (1980). The main purpose of this paper is to establish (1.1) for a strictly stationary strong mixing sequence under somewhat less restrictive conditions. We should point out that our results clearly contain Yokoyama’s results when g > 4. For 2 < g G 4, our results are better only when the existence of moments of Xi is restricted to some low order. See remark after Theorem 1. We shall show how Doob’s argument (see Doob, 1953, pp. 225-227), which is the basic tool for Ibragimov’s proof for &mixing, can be applied to strong mixing case in a straightforward manner. In Section 2, we discuss the mixing conditions and basic moment inequalities. In Section 3, the main results are stated and discussed and their proofs are given in Section 4. Section 5 presents some applications of our results.

2. Dependent random variables Let {Xi} be a strictly stationary +-mixing or strong mixing sequence. Under &mixing,

‘p(AnB)-P(A)P(B)‘;AESk BEC+,,k&l

sup

=4(n)JO,

1 3

P(A)

i

I

whereas under a-mixing (strong mixing), sup(lP(AnB)

-P(A)P(B)I;

A&F:,

BET+,,,

k>l}

=a(n)JO

holds where s”,” is the a-field generated by {Xi: a G i Q b}. For such mixing conditions, the following two basic inequalities are well known. Let 5 and 17 be measurable with respect to F/ and G+,, respectively. Then, under &mixing, lJ%r?)

-~~5~~~77~I~~~~~~~11’Pl1511Pl17711q

(2.1)

where 1 QP, q d 03 with l/p + l/q = 1, and under a-mixing, lE(577) [email protected])E(n)I< where 1

12[+)11-(1’p+1’q)l1511pl17711q

q < m with l/p + l/q

(2.2)

< 1. For their proofs, we refer to Ibragimov (1962) and Davydov

3. Main results

Theorem 1. Suppose {Xi} is a strictly stationary sequence of random variables satisfying the strong mixing condition. Assume EXi = 0 and E I Xi I g < m for some g > 2. Zf 5

[‘y(i)]‘g-2)‘g<03

and

tilfh[a(i)]l-h

i=l

(3.1)

for some 0 < h < 1, then there exists a constant K such that

n=l,2

164

)....

(3.2)

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Theorem 2. Suppose {Xi} is a strictly stationary sequence of random variables satisfying the strong mixing condition. Assume EXi = 0 and 1Xi 1 Q C. Zf i~li”“[cx(i)]‘”

(3.3)

for some 0 < h < 1, then (3.2) holds. Remark 1. For (3.2) to hold, Yokoyama’s (1980) Theorem

1 requires

E 1Xi 1g+” < ~4,

$i+

1)

g/2-1[ a( i)] u/‘g+“’

for u>Oand

g>2.

(3.4)

If g > 4, then ig - 1 > 1 and u/(g + U) < 1. Since we can choose 0

should point out that if 2

2 (i + l)g’2-1

to

strong mixing process Yokoyama’s Theorem 2 requires [a(i)]

for g>2.

i=l

It is clear that Yokoyama’s Theorem 2 is more restrictive than (3.3) for g > 4.

4. Proof of theorems 1. It is readily seen (see also Theorem

Proof of Theorem

1.7 of Ibragimov, 1962) that,

2

E 5X, I i=l

(4.1)

I

if E I Xi I g < CQand C~=D=I[~(i)l (g- 2)/g < w . This shows that (3.2) if true for the second moment under the conditions of the theorem. It will therefore be sufficient to prove that (3.2) is true if g is an integer m > 2 and prove that it is then true if g = m + v for 0 < v G 1. We thus assume in the following that (3.2) is true if g = m. Define S,, T,, S,, C, by n+B, s,=

tx,, i=l

Tn =

C i=n+l

Xi,

c i=n+B,+l

where B, is an increasing sequence of n, to be determined C, < fi(m+cV*

F?l+l.

2n+B,, $=

Xi

and

C, = E 2X, I i=l

I

more precisely below. Then we are to prove (4.2) 165

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choice of K. In order

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27 January

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to prove this we first prove that if &i > 0,

E 1S, + $, 1m+”< (2 + &i)C, + zqn(m+~‘)‘2, for a proper

choice of K, and B,. In fact, remembering

E 1S, + in 1m+L’

that S, and S,, have the same distribution,

[ini”)

Now

Pr(I S, I f > GE

Cr/(m+c’)[ n

a(

-t/(m+c+s))

B,)]

-(m+ir)/(m+u+d))

““+u’,(c,[a(B,)]

( sn1

<[(Y 4Jl

(m+l;)/h+u+s)

for some s > 0 and 0 < t < m. Taking

B, = n”/(“‘+“) for $(m + u)

(m+u)/G?l+u+s) = g1 [a(n’/(n2+c))](m+“)l(m+a+~). Since

m m

a

Xr/(m

fd)]

(m+l~)/h+lJ+S)

dx

<

o.

iff

1

2 [ cu(n’/(“‘+U))](m+c)/(m+v+s)

< co

n=l

and

(m+L’)/(m+L.+S) dx it

=

(m

+

“),‘llmy,““‘-‘),‘[a(y),(“‘+‘:‘/‘m+a+j)

dy,

can be easily seen that [ (y(n’/(m+“))](m+~)/(“+“fS)

C

< 03 iff

n=l

wherel<(m+u-r)/r<2andO<(m+v)/(m+v+s)

c

n(l+h)[ a(

n)]“-h) <

to change

the right-hand

side in the last expression

to

(4.3)

00,

n=l

for some 0

close to i(rn

+ U> and

1S, 1’ < C;,(m+L.)[ a( B,)] --f/(m+“+s) and

1 for all sufficiently large n (n a Nl) and 0 < t =Gm by the Borel-Cantelli depend on t on account of (4.3).

w.p.

166

Lemma.

Here,

N, does not

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When j = 1,. . . , m, apply (2.2) with p = ~0, q = m to obtain E 1S, 1j 1in 1m+“-i

-i/(m+u+s)C~,+,-I)/(,+I:)[(y(Bn)l

< [EIS,I’“]““[EIS,,I”]‘“+“-“‘“+

-(m+c-j)/(m+E,+S)

12C,[a(B,)]“““+“+“’

+ 12C,[cX( B,)]S’(m+c+S),

for all n > N,. Here we are supposing (4.2) is true if u = 0. When j = 0,. . . , m - 1, a similar argument leads to EIS,Ii+“I$lm-j

+ 12C,[a(B,)]S’(“+~+?

Combining these results, we find that when n > N,, EIS,+in,m+c

,< (2+b[u(B,)]3’(m+U+S))C,+KIIE(“+“)’*

for some constants K, and b not involving n. Then (m+o)

2n+B,

C,, = E S, +$

+ T, -

c

X,

i=2n+l n+B, <

El/(m+U)

1 S,

+ in

I (m+o)

+

+

i i=n+l

I

h+a)

2n+B,

c

c i=2n+l

El/(“J+c)

1 Xi

1 cm+“)

1

I

h+c) <

2 +b[a(B,),“/‘“+“+“‘)C,

+K,n”“+“‘/2)“‘m+i”+

2B,Cf”“‘+“]

.

[((

Increase 12so that b[a(B,)]“/‘“+““’ [{

< E, (if necessary) and then the above quantity is bounded by

(2 + EI)C, [email protected]&m+l:)/2)r/(~+~‘) + 2B,C:/““+“‘](m+U) 3

for all sufficiently large 12(n > N,). Since B, = o(n’/*), < ((2 + &i)C, + K,n(m+U)‘2}

l/h+l,)(l

the above expression is

+ El,](-~)

[

for sufficiently large n (n 2 N3). Finally we get c,, G (1 + Ei)(m+L.)[(2 + &i)C, + K,n(“+“)‘*] for all n 3 N3. If &I is so small, (1 + &i)@+“)(2 + E,) =G2 + F. Thus there must be K, for which C,, G (2 + &)C, + K,~z(“~“)/~

for II > 1.

(4.4) Now by following the argument on p. 227 in Doob (1953, proof of Lemma 7.4), we complete the proof of (3.2). q Proof of Theorem 2. It is easy to check that

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if I X, 1 G C and &o(i) 0

5. Application

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21 January

1993

< ~0 which replaces the first part of (3.1). Then under (3.3), the proof follows.

of results

In this section, we state some limiting laws for strong mixing random variables. The detailed proofs of the following results may be found in Yokoyama (1980). Theorem 3. Suppose that the assumptions of Theorem 1 or 2 hold. Then, as n + m,

S,/[h(log

n)“g(log

log n)2’g] + 0

as.

q

Theorem 4. Suppose that the assumptions of Theorem 1 or 2 hold. Then there exists a constant K such that

Et

k=y~

,

n

IS,

tg)GKrP2.

0

9

Theorem 5. Let (Xi1 be a strong miring sequence. Zf the assumptions of Theorem 1 are satisfied, then as n -+ m, El S,/(oh>

lg + P,

where ~3, is the gth moment of N(0, 1) and o2 = EX:

(5.1) + 2C7=, EXIXj.

0

Theorem 6. Let {Xi) be a strong mixing sequence. Zf the assumptions of Theorem 2 are satisfied, then as n --f m, (5.1) holds. q

Acknowledgement

I am grateful to an unknown referee for some useful comments and the Professors at Department Statistics, Keimyung University for their support.

of

References Bahr, B. von (19651, On the convergence of moments in the central limit theorem, Ann. Math. Stat&. 36, 808-818. Brillinger, D. (19621, A note on the rate of convergence of mean, Biometrika 49, 574-516. Davydov, Yu.A. (19681, Convergence of distributions generated by stationary stochastic processes, Theory Probab. Appl. 13, 691-696. Doob, J.L. (1953), Stochastic Processes (Wiley, New York).

168

Ibragimov, I.A. (19621, Some linit theorems for stationary processes, Theory Probab. Appl. 7, 349-382. Ibragimov, LA. and Yu.V. Linnik (19711, Independent and Stationary Sequences of Random Variables (WoltersNoordhoff, Groningen). Yokoyama, R. (19801, Moment bounds for stationary mixing sequences, 2. Wahrsch. Verw. Gebiete 52, 45-57.