79
Internat. J. Math. & Math. Sci.
VOL. 21 NO. 4 (1998) 749-754
COMPLETE CONVERGENCE FOR B-VALUED L’-MIXINGALE SEQUENCES
UANG HANYING and REN YAOFENG
Department of Statistics and Finance
University of Science and Technology of China
Hefci,Anhui 430072, P.R.China
and
The Naval Academic Institute of Engineering of China
Wuhan 430033, P.R. China
(Received April 19, 1996 and in revised form January i0, 1997)
ABSTRACT. Under weaker conditions of probability,we discuss in this paper the complete convergence for the partial sums and the randomly indexed partial sums of B-valued /)’-mixingale
sequences.
KEY WORDS AND PHRASES" Complete convergence,LV-mixingale sequence, q-smooth
Ianach space.
1991 AMS SUBJECT
1.
CLASSIFICATION CODES: 60F1560B12.
INTRODUCTION AND MAIN RESULTS
Since the definition of complete convergence for real random variables was introduced by ttsu
and Robbins Ill,there have been an extensive literature in the complete convergence for independent and dependent random sequences,see partially the references listed.In particular,Yang[5,6
ha. discussed the complete convergence for B-valued independent random elements.Yu[ll] ha
co,sidered the complete convergence for martingale difference sequences, PeligradI7 and Shao[S}
have obtained the complete convergence for C-mixing sequences,respectively.tlowever,to our best
acknowledgement,there are still few articles on the complete convergence for L-mixingale (1 _<:
p < 2) sequences,which include uniformly rnixing (called also -mix;ing) sequences,martingale
difference sequences,linear processes and other random sequences (see I0]).In this paper,under
wakcr conditions of probability,we discuss the complete convergence for the partial sums and
the randomly indexed partial sums of B-valued L-mixingale sequences, and give the complete
convergence for B-valued martingale difference sequences as corollary.The methods used here are
different from those used in the literature.
Next,let us introduce some notations.
Let B be a real Banach space.B is said to be q-smooth(1 < q _< 2) if there exists a constant
>
C 0 such that for every B-valued L -integrable martingale difference sequence (D," > 1
Ell
EIID, ", , >
,.
be a sequence of B-valued L -intcgrable (1 < p < 2) random variables on
probability space (I,.,P), and let {Y’,,-cx < n < o} be an increasing sequence of sub afield of Y.Then {X,, Y,} is called a LV-mixingale sequence if there exist sequences of nonnegative
constants C, and (n),where (rn) 0 as m
ov ,which satisfy following properties"
lct
a
{X,,n > l}
D, II’ < C,
50
L. HANYIIqG AND R. YAOFENG
(i) IIE(XIT_.,)I, <_ (m)c and
(ii) IIx,,- (x,,l:r,,,,,,)ll, _< O(,r, +
for all n > and m > O,where IlXll, (EIIXll) ’/".
S is the class of all positive non-decreasing function
satisfying the following conditions:
(i) There exists a constant, k k() > 0 such that
on
R+
[0, o)(see [9],p.228 or [5,6])
(=v) _< ((=) + ()), v=, v n +.
-
is non-decreasing for sufficiently large x.
From now on ,we will use C to denote finite positive constants whose value may change from
statement to statement .For real number z, Ixl denote the largest integer k _< x.I(A) represent
indicative function of set A. Put S
X,.
(ii) z/qb(z)
THeOreM 1.1. bet
< q2,0< <
1,1 p 2, d= or-l,and let B be a
q-smooth Banaeh space.Supppose {X, } is a B-valued L-mixingale sequenee,() S.If
e(llX, ll’(e(llX, ll))
for sufficiently large x,n and there exists a
X(1
-
> =)
A
<
q-t
for every
(i,then
ii
-
lp( m
in particular
n=l
/2
(.)
p) such that + (1
mx C
([n’l) l<t<n
where 0 <
c=-’’
t)A > 0 and
(1.2)
<
> 0 we have
IISll > (n(())) ’/’) <
P(IIS.II _> (,((,))’)’/’) <
(.3)
(1.4)
oo.
THEOREM 1.2. Under the assumptions of THEOIEM 1.l,if there exists
satisfying
m2([2"])
then for every
>
0 we have
max
-1p(up[ll&ll/([(k)))’/’
a
A(1
C<,
)<
A
p)
(1.5)
.
(1.6)
n=l
{X,,,n 1} is a -integrable
(EIIXII) v, (m) 0 for m 1.
If
COROLLARY 1.1.
space.Supppe {X,, ,,n
<
B-valued martingale difference sequence,then
C
Let
or-l,and let B be a q-smooth Banach
< q 2,d
a
S.For 0 <
B-valued
is
difference sequence,(z)
martingale
1}
and suciently large z and n,if (1.1) is satisfied,then for every > 0, we obtain that
( .z),(.4) ,a (.6)
RMARK 1.1. For 0 < < 1,by C-inequality and properties of (z), we can prove
THEOREM 1.1,THEOREM 1.2 and COROLLARY 1.1 hold for y B-valued
hal, the results of
rando,n variable sequence
{X,,n
1}
wihou mixing condition
(1.2)
REMARK 1.2. Real uniformly mixing sequence (definition
L -mixingale,where C, 2(EX)/,(m)= /(m),see [0,p.9].
see
and
[7]
(1.5).
or
[s],[x01) (x, ,}
i
COMPLETE CONVERGENCE FOR THE PARTIAL SUMS
751
REMARK 1.3. YangI51 has proved that (1.3) and (1.4) hoM for B-valued independent
zero mean random element sequence {X.} in type 2 Banach space under moment conditions
stronger than the conditions of COROLLARY 1.1.
2. PROOFS OF MAIN RESULTS
We only prove the case in d
for Theorem 1.1 and 1.2,the proof of the case in d -1 is
analogous.
LEMMA 2.1.([9],Lemma 1) Let (.) E S, 6 > 0,then for any z > 0,
c(=) < (=(=)) <
c() <_ (/()) < cc);
c() < () _< c().
PROOF OF THEOREM 1.1. Notice first
Obviously
-P(max
IISll >
l<k<rt
_< .=,E -P(,<<.max IIA,II >_ (=(=))’/’)
n=l
’
;p(,<m<x. IIcll _> (())’/’)
.:,
g I+Is+13.
the Markovian inequality,/)’-mixingale property and the properties of
,
_< c
<_ c
n=l
(x),we have
(,,(.//-#(
(())-/’(: IIX,- ECX, l,+[..])ll)
=1
max
C,
"-1
<C 2
rt=l
max
(InoI) l<<n
Similarly,we can obtain
13 < C
max C < oo.
(lntl) l_<,<n
< I,i
Clearly,X,
Y,,. + Z,,.,Wt,
U,, + V,,.For
fixed t, (U,,E+,
_< _< n} and {V,,,.,+t, <
752
L. HANYING AND R. YAOFENG
__< n}
are martingale difference sequences. Then
l=-[nPl+l
l<k<n
t=l
g I + I.
Since B is q-smoothable,therefore using Doob inequality, the monotone property
and iemma 2.1 we have
n=l
l=-[nP]+
t=l
n=i
n=l
=1
By applying the definition of (x) and Lemma 2.1,we can obtain
e( )
ce
for ,/ > 0 and sufficiently large z.
By the Markovian inequality, the definition of $(z),Lemma 2.1 and
(2.)
(2.1)
we have
The proof is completed.
PROOF OF THEOREM 1.2. First,by the monotone property of (x) we have
n=l
>__
1’(sup(ll&lll(k(k))’/’)
k>n
P(sup(llSll/(k(k))’/’) >_
<_
,=0
< C E raP( max
2"5k<2
m=l
Observe that for 2
(llS, >
<_ k < 2 "+,
,s’
E(x,t"-I
+
iv""l
E
E (E(x, 13;,+,)-
=-12""1+1 ,=’
+E
COMPLETE CONVERGENCE FOR THE PARTIAL SUMS
753
Then
kpCsup(lls,
ll/((,)) ’#) >_ )
k>n
n=l
< C
m=l
max (IIAII >
mP( m+
E raP(
(2())’/’)
(IIBII >5
+c mP(.. <<-+,max ([[c[[ >_ 5’(2m(2m)) ’/’)
I+Ir+I.
proof of I we have
+C
max
f
fly analogizing the
m:Z’f([2’])
I6 _< C
max
C, <
oo.
m=l
By analogizing the proof of Is,similarly, we can obtain 18 < oo.
Let Y,.,, X,/(llX, <_ (2"’(2"))’/t),Z,.,, X,-Y,.,.,,,W., E(X,[+,)-E(XIT,+,_,),U,,E(Y,,.IZ+,)- E(Y,.,IZ+,_,),V.,--- E(Z,,..IZ+,)- E(Z,,,.,,IZ+,_,),2" < k < 2"+’,, < <
k, -121 +, _< _< [2a’l. Then
I7 < C m=l m
+C
E
E
I=-[2"]41
E
m
m=l
max
P(.. <k<2m+t
l=-[2a’]+l
*=1
max
P( 2" <k<
2"4
E v,,,ll > (2"(’)) ’/’- 2 -)
t=l
19 + 1,0.
ly analogizing the proof of
19 _<_ C
we have
m2(a+qa+’-q/O((2"’))-q/t + C
llv analogizing the pr-of of
1,o <__ C
h
Is
m2 (--’)"
<
.
we have
m2 (-’)
m2 ("-‘)
+C
j,[((yt)),+t
RANDOMLY INDEXED PARTIAL SUMS
Throughout this section let {X,,7,} be a B-valued LP-mixingale sequence (l
p
2),
and let {r,, n 1} be a sequence of nonnegative,integer valued random variables.r is a positive
random variable.All random variables are defined on the same probaSility space.
THEOREM 8.1. Under he sumptions of THEOREM 1.2,if there exists some constant
to > 0 such that
S.
then for every
> 0 we have
.,
,
-e(-- < ’0)< ,
(3.1)
n=l
1p(llS,.I
rt
>_
,(.((.)))’/’)
<
oo.
(3.2)
THEOREM 8.2. Under the assumptions of THEOREM 1.1,if there exist constants a, b, t0(0 <
a_<b<oo) such that P(a _< r _< b)
n=l
then for every
>
0 we obtain that
and
P(I-- *1 > ,0) <
(3.2)
holds.
754
L. HANYING AND R. YAOFENG
THEOREM 3.3. Under the assumptions o[ THEOREM 1.l,i[ there exist constants b > 0
and (3.3) is satisfied,then for every > 0 we have
P(r _< b)
and e0 > 0 such that
n=l
Obviously,suppose
P(r > a)
> (((,,))’)’/’) <
[P(llS,.,,ll
n
1 for some a > 0,then for any
p(r,,
<
a
> 0(e < a)
we have
e) .<_P(l
for some a > e0 > 0 replaces condition (3.1},then
therefore,if condition (3.3),where P(r _> a)
TIIEOREM 3.1 still holds.
Similarly,using COROLLARY l.l,we can obtain the complete convergence for the randomly
indexed partial sums of B-valued martingale difference sequences,respectively.
REMARK 3.1. Condition (3.1) and (3.3) are just ones which are usually employed in
literature.
REMARK 3.2. Note that if (x) 1,Lk(x)(Lo(x) max(1,1og x),L log[max(e, Lk_,(x)],
k
1,2,.--),we can derive many significative results from the results of this paper.In addition,
since real space is a 2-smooth Banach space,the TttEOREM and COROLLARY in this paper are
sitable for real valued random variable.
ACKNOWLEDGMENTS. The authors would like to thank Prof.Hu Dihe for his suggestions
and encouragement.In addition,they are very grateful to the referee and the editor for valuable
suggestions and comments.
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