Complete convergence for α-mixing sequences

1) be a strictly stationary a-mixing sequence of random rsariables with EX, = 0, (E 1X, 1r)‘/r < 00. Assume that

IE n=l

cu’/O(n)

Correspondence to: Qi-Man Research

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< m

for some f3>

Shao, Department

by the Fok Yingtung 0 1993 - Elsevier

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of Mathematics,

Education Science

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Then

(l-2) However, a contrary example to Hipp’s conclusion was given by Berbee (1987) when r = 03, i.e., in the case of I X, I bounded. The aim of this note is to discuss whether Theorem A is true or not in the case of r < 03. Throughout this note we will use the following notations: S, = Cy=,Xi; [xl denotes the integer part of x, I{Aj the indicator function of the set A and log x = log, maxk, 21, the logarithm with base 2; x xy means x = O(y) and y = O(x); x A y = min(x, y); and IIX IIr denotes (E I X I r)l/r. 1, 1 < l/cu

1) be an a-mixing sequenceof random m-iables II X,, III < w. Assume that with EX,, = 0, sup,, ~, Theorem

4<

1. Let

a(n)

or <

= O(n

--r(p--l)/(r--p)

log-P

for some p > rp/(r

II)

-p).

(1.3)

Then, (1.2) holds true.

An immediate consequence

of Theorem

1 with p = (Y = 1 is:

1. Let 1 < r G cc),IX,,, n 2 11 be an a-mixing sequence of random variables with EX,, = 0, sup, BJ X,, (Ir < 00. Assume that

Corollary

cw(n)=O(logPPn)

forsomep>r/(r-1).

Then

5 qyy

n=ln

aen

.

) < cQ for all F > 0.

In particular, we have S,/n

-+ 0

a.s.

0

This corollary does not remain true if p > r/(r - 1) is replaced by p > r/(r - 1). 2. Preliminary

inequalities

To prove our theorem, we need the following lemmas, which are of independent

interest.

Lemma 1. Let (X,,, n > l} be a sequence of random variables with EX,, = 0 for every n z 1. Then

P(y$ty

BX) <4x-’

5 E I Xi 1I{ 1Xi I > c} + 4x-”

+ (32)3ncx-‘a(

k)

(2.1)

i=l

for any a >, 1, x > 1, c > 0 and integer k satisfying

1 G k
(2.2)

log x)

and for some s > 2, ( i i=l

280

I\XiI{ ( Xi 1
i i=O

alP2/‘(i)

Gx2/((32)3a

log x).

(2.3)

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Proof. Let

X,=X;Z{IX;I
a),

1 1 3,=j=l i x,, T,, T.J =ji=0y.2. =ji=0q.19 (Zi+l)kAn

r,,,=

c

X,,

i=O,l,...,

q,:=

L-i,

[

,=1+2ik

2k

2c1+ INAn c

x,2 =

xj,

I +(2i+

j=

&

i=O,l,...,q,:=

-1

)

[

1)/i

It is easy to see that max

i
I S, I + i

I S; I < max

i <,I

i=

I Xi I I{ I Xi I > C}+ f E I Xi I ‘{ I Xr 1> ~1 i=l

I

and

P(maxISjl 2s)

2;~

1

i < II

i
eE(X,II((Xjj

+4x-’

(2.4)

>c}.

i=l

Since max I 3, I G max

I + max I T,,z I + 2kc G max I T,, I + max I q,2 I + 4x, lki
I T,.,

O
IQU

by (2.21, we have

(2.5) We first estimate

Z,. The estimation

G- ,=u(fl),

of I1 is completely

G,=cr(Xj,

l
similar.

Put

ui=Y,,,--E(y,,

IGj_l),

q.=

cuj, j = 0

i=o,

l,....

Then max O
and {U,, G;, i > 0) is a martingale

E( e’“f IG,_,)=l+

with

I U, I 2 &x

1

:= Z, + Z4

I u, I < 2kc for every i > 0. Noting

(2.6)

that for each real t and i > 1,

?E I=2 =

< 1 + t*E(uf G

I G;_,) c

exp(t’e21f1”“E(uT

(ltl2kc)’

I=0

I, .

I G,_,))

i exp( t2e2”‘kCE( yf, I Gjp ,)), 281

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we find that

exp i

tU, - t2e21rikc k

E(qfl

IGj-1)

j=O

i

is a non-negative

for every t and hence

supermartingale

P

tLJ-

t2e2”lkc

k

E(l$f,

IG,_,)

(2.7)

G l/Y

j = 0

for y > 0, by the maximum and (2.2), we have P

inequality

max U;>&x ( O
(cf. Stout,

1974, p. 299). Take

t = (32a log x)/x

in (2.7). By (2.7)

1 i

tU, - t2e21’ikC c E( ?;I 1G,-,) j =0


> exp kxt i

- 1 ’ e 2i’1kc,ii,E(E;:

lG,-,))I

GP t2e21tIkcX2

i +

P i

max exp tUi - t2e21tikc C E(Y,; O
I Gj-,)

a

exp

-&xt+

>

4(32)2a

Gj_ I) Using the well-known

Davydov /

\

/ k

X2

’ 8(32)2a 282

>

log x

log x

4(32)2a

log x

X2

(2.8)

+X-a. 4(32)2a

inequality,

k

log x

t2e2i’IkcX2

X2 1)

4(32)2a

i

i

Gj-

&xt -

log x

one can obtain 41

(2j+

I)k

(cf. Davydov,

1970, and Moricz,

1985)

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by (2.3). Hence

X2

9I

P

i

c

E(Y,f,IG;_,)>

4(32)2a

j=o

t((32)‘a

log x 91

<

C+(Y,: j= 0

X2

Write

log x

I&)-%

I G,__ ,) - Ey.,<. We find that

5, = E(y.,: El,$l

=E(Y;T,

by the Davydov

inequality

-ET;)

again.

(2.10)

sgn ,.$,~4(kc)~a(k) Inserting

(2.10) into (2.9), we obtain

‘1I P

(2.9)

1.

C E(Y,~,

IGj-,)~

i j=O by (2.2). Now a combination

X2 4(32)2a log x

~

(32)“anc%CI(

k)log

(32)2nc(u(

X

k) (2.11)

< X2

X

of (2.8) and (2.11) yields

P

Similarly,

we have P

max U, < - Ax
+ (32)‘ncx-‘cu(

k).

Hence I, < 2x-“ Similarly

(2.12)

+ 2(32)2ncx-‘a(k).

to (2.10), one can get that E(E(Y,,,

IG,_,)I=EK,,

sgn(E(K,,

lG,_,))~4kca(k)

and hence (2.13)

Z,<64ncxY’cr(k). It follows from (2.6), (2.12) and (2.13) that I, < 2x-” Similarly,

+ 3(32)2ncx-‘~(

k).

(2.14)

k).

(2.15)

we also have I2 < 2x-”

+ 3(32)2ncx-‘a(

This proves (2.1) by (2.4), (2.5), (2.14) and (2.15). Lemma

2. Let {X,,, n > 1) be an a-mixing sequence EX, = 0,

IIX, II1,
and

0 of random

cx( i) < C,ip’y

rjariables. Assume

log-”

that

i 2x3

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for i > 1 and for SOme CC2 v > 1, C, > 1, 7 > 0 and real A. Then there exists a finite positive constant K depending only on V, 7, h and C, such that .(,,ISil

ax)

for every x >KDn1/2

(2.16)

~Kn(D/x)“(‘+‘)““+‘)log(“-‘X’-“)‘(’+”)(x/D)

log1+1”1/2 n.

proof. We assume, w.l.o.g., that D = 1. Otherwise, constant K such that P( max 1Si 1 ax)

< fi-u(T+w(~+T)

put X,’ = Xi/D.

log(V-‘XT-w(~+~)

It suffices to show that there

x

exists a

(2.17)

i=sn

for every x >JGz’/~

n. Take

log1+lAl/2

c = 2x7/(7+y) log(h-7MVfT) in Lemma

1. Assume

x,

a=T+2,

k = [ x/(64ac

log x)]

that

&&.-v(7+i’)/(v+7)

log(V-lHT-~M~+4

x < 1.

(2.18)

Otherwise, (2.17) is trivial. If the conditions of Lemma 1 are satisfied, then (2.17) will follow from (2.1) immediately. So we only need to verify (2.3) satisfied. In what follows we denote K, the finite positive constant depending only on V, 7, A and C,, whose value may be different from line to line. If 1 < u < 2, then ( e

I xj I < C}li$) i alp2j2(i)

]Ixil(

i=O

i=l G

2nkc2-’

< xnc’-“/(32a

X‘

=

(32)3a

log x)

. (32)221-unx-“(~+l)/(“+‘)

~og(v-lX~-AM~+7)

x

log x

X2

’

(32)3a

v > 2, we have

by (2.18). When

i

(2.19) log x

i$lIIxil{ I xi I GC}il:) 2a1p2/v(i) i=O /
k

1 G

\

1 + C, C i-T(1-2/y) i=l

K,n(log

284

(32)3a

+

.

( K,rK2

k-T(l-2/u)+l

log-“(‘-2/V’

log x))-7(1-2’Y)+1 log 2+lAl(l--2/v)

x

k)

log-A(1-2/v) +~lm-“(‘+l)/(“+‘)

x) 10g(Y-lXT-h)/(V+7)

x)

log x

XL

’

k

x + (x/(c

X2

(32)3a

i J

1+lhj(1-2/v)


’

log-“(1-2/“)

(2.20) log x

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& PROBABILITY

by (2.18) and x 2 Kn1/2 log’+1”1’* n. Now we conclude completes the proof of the lemma. •I Proof of Theorem that PC

1. By Lemma

max

I Si I 2 &rP

1993

from (2.19) and (2.20) that (2.3) is satisfied.

2, we have that for every E > 0, there

Q Kn’-“‘(‘+“P-I)/(r-p))/(r+r(P-I)/(r-p)) log-

16 March

exists a positive

constant

This

K such

1

ign

= fi’-pa

LETTERS

log (1 -rKPr(P-

1 -(r-PXP-rP/(r-P))/r

which yields (1.2) immediately

n

I)/(r~P))/(r+eP-

l)/(rpP))

n

7

q

by (1.3), as desired.

3. Examples The example p >r/(r-P). Example

below shows that Theorem

1. Let r > p > 1, 1 > (Y2 l/p.

a(r -P>

a=

r( 1 - a) +pa

g(x)

=xa logd

G(0)

= 0,

x,

valid, if the assumption

of (1.3) is replaced

by

Put

b=

- 1’

1 does not remain

a(r, - 1)

,

-1

d=

r-1-a(r-p)’

r-p

x > 0,

G(n)

= 5

II=

[s(i)]>

1,2,...,

i-1

f(x) = (g(x))

g(X))r’(r-p)

T(p-‘)‘+-P)(log

Let (Y,, n > l} be an independent

sequence

log log g(x),

x>o.

of r.v.‘s with 1

P(Yn = &p”(n)) Define a sequence properties.

EJX,

lr= 1,

= O(n-‘(P-‘)/(‘-P)

f

f(n) .

{X,, n 2 1) by X, = y for G(j - 1)
EX,,=O, a(n)

P(y,=O)=l--

= &,

nPap2P(

Then

{X,, n 2 1) has the following

(3.1)

log-~A-P)

fl log-’

IS, ( > .5yta) = 00 forall

log n),

E>O.

(3.2) (3.3)

PI=1

Proof. From the above definitions, a+b=(l+a)cu, ar(p r-p

- 1)

we have the following 1 r-p

-(pa-2)(a+l)-a=2,

+ d(P

- 1)

r-p

relations

+d=da,

(3.4) (3.5)

285

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4

r

+ dr(p-

r-p f(n) =n f(g-‘(,2)) G(n)

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16 March

1)

-(pa-l)d=l, r-p ~eP--1)/(1.-P) log r/(r-/J)+dr(P,” n~(P-lv(r-P)

xn”+’

LETTERS

1993

(3.6) l)/(r-P) n log log n,

logr/(r-/dn

(3.7)

log log pl,

(3.8) (3.9)

log“ n.

construction of {X,,, n > 11, we find that for ~1, k a 1, IX,, . . . , X,) and are independent, unless there exists j such that G( j - 1) < k, n + k < G(j). In this {X,1+& X,l+k+lY.J case we obtain by an application of Lemma 8 of Bradley (1981) and Lemma 2.1 of Herrndorf (1983), According

to

the

+(X1,...,

X,)7

U(Xt,+k,

)) =a(a(q),

Xn+k+l,...

G l/f(j).

a(q))

Hence a(n)

[g(j)1 z=n+ 1)G l/f(g-l(n)).

Gsw{l/f(j):

This proves (3.2) by (3.81, as desired We next check (3.3). Put 7;=

C [g(i)]I$,

j=

1,2,...

r=l

For G(j) symmetric

< n < G( j + l), we can write random variables, we have

S,, = T, + (n - G( j>)y+ ,. Since

{y, i a 1) are

independent

)afP(max[fi(i)]l~I~ZFC;“(j+l)).

P( IS,, I a EP) a +P

(3.10)

Let mj=min{i:

[g(i)]f”“(i)a2cG”(j+l)}.

From (3.4) and (3.7) it is not difficult

to see that mj < ij for every j sufficiently

large. Therefore

j ‘K. for some positive

constant

1)

f(j)

K and for every j sufficiently

large. By (3.10) and (3.11), we conclude

G(j+ I)

c /=I

k”“-2

>K

min G(j)
k=l+G(j)

P( I)

G”“-2 (j)g(j)j/f(j)

f j=l

1

>Kf j=

,

j log j log log j

=CC

by (3.4), (3.7), (3.9), (3.5) and (3.6). This completes 286

the proof of (3.3).

q

I Sn

/ >

EdY)

that

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2. Let r > 1. Put a = (r - 1)/r,

g(n)

= [rF’

exp(n’)],

G(n)

= i

g(i).

i=l

Let {Y,, n > 1) be an independent

sequence

of r.v.‘s with 1

P(Yn = *n”r Define

logI”

n) =

2n log n ’

X,, = Y. for G( j - 1) < n < G(j). EX,=O,

a(n)

E(X,

I’=

= 0(1og-“(r-n

1

Then

P(Yn=O)=l-p

II log n .

{X,, n 2 11 has the following

properties.

1,

log-’

log n), for all E > 0.

Proof. The proof is exactly similar

to that of Example

1 and so is omitted

here.

0

We now turn to the correctness of Theorem A. Assume 1

8 provided p n large enough. But our Example 1 says that the mixing rate is required at least n -(p~1)/2. This means that Theorem A is quite possible not true. Unfortunately, {X,, n & 11 in our Example 1 is not strictly stationary. We conjecture that there is a strictly stationary a-mixing sequence that the {X,, II > l} satisfying (3.1), (3.3) and a(n) = O(n~‘(p-‘)/(‘~p) log-’ n). We also conjecture assumption p > rp/(r -p> in Theorem 1 can be replaced by p > T/(Y -p>.

Acknowledgement The author

wishes to express

his gratitude

to the referee

for his valuable

suggestions.

References Berbee, H.C.P. (19871, Convergence rates in the strong law for bounded mixing sequences, Probab. Theory Rel. Fields 74, 255-270. Bradley, R.C. (19811, Central limit theorems under weak dependence, J. Multivariate Anal. 11, 1-16. Davydov, Yu.A. (1970), The invariance principle for certain probability limit theorems, Theory Probab. Appl. 15, 487498. Hermdorf, N. (1985), A functional central limit theorem for strong mixing sequences of random variables, 2. Wahrsch. Verw. Gebiete 69, 541-550. Hipp, C. (1979), Convergence rates of the strong law for stationary mixing sequences, Z. Wahrsch. Verw. Gebiete 49, 49-62. Hsu, P.L. and H. Robbins (1947), Complete convergence and the law of large numbers, Proc. Nat. Acad. Sci. U.S.A. 33(2), 25-31.

Lai, T.L. (1977), Convergence rates and r-quick versions of strong law for stationary mixing sequences, Ann. Probab. 5, 693-706. Moricz, F. (1985), SLLN and convergence rates for nearly orthogonal sequences of random variables, Proc. Amer. Math. Sot. 95, 287-294. Peligrad, M. (1989), The r-quick version of the strong law for stationary $-mixing sequences, in: G.A. Edgar and L. Sucheston, eds., Almost Everywhere Concergence (Academic Press, New York) pp. 335-348. Shao, Q.-M. (1988), A moment inequality and its applications, Acta. Math. Sinica 31, 736-747. [In Chinese.] Shao, Q.-M. (1989), On complete convergence for p-mixing sequences, Acta Mafh. Sinica 32, 377-393. [In Chinese.] Stout, W.F. (1974), Almost Sure Convergence (Academic Press, New York).

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