Convergence rates in the law of large numbers for B-valued random elements

2001,21B(2):229-236

CONVERGENCE RATES IN THE LAW OF LARGE NUMBERS FOR B-VALUED RANDOM ELEMENTS 1

*)

Liang Hanying ( :ifh'~. Wang Li ( .I..ijij ) Department of Applied Mathematics, Tongji University, Shanghai 200092, China Abstract

The author discusses necessary and sufficient conditions of the complete con-

vergence for sums of B-valued independent but not necessarily identically distributed r.v.'s in Banach space of type p, and obtains characterization of Banach space of type p in terms of the complete convergence. A series of classical results on iid real valued r.v.'s are extended. As application authors give the analogous results for randomly indexed sums. Key words Convergence rate, random element, Banach space of type p, slowly varying function 1991 MR Subject Classification

1

60B12

Introduction and Main Results Let {X n , n

put S" =

P(ISnl/n

~

" LX;,

n

;=1 1 t /

I} be a sequence of random variables in the same probability space and

>

1. In real-valued case, the rates of convergence to zero of quantities

> f) (0 < t <

2) are described by the well-known theorem of Baum and Katz

(1965): Theorem BK Let 0 1::::; t < 2, EX 1 = O. Then

< t < 2,r

~

1, and let {Xn,n

00

Lnr-2p(IS"I'~ f.n 1 / t) < +00 "If>

0

~

I} be real valued iid r.v.'s, when

<===> E!X 1 1rt < +00.

,,=1

In 1985, Bai and Su generalized Theorem BK and obtained the corresponding results of randomly indexed sums. The aim of this paper is to extend the results of Bai and Su (1985) to not necessarily identically distributed r.v.'s in Banach space; to find its necessary and sufficient conditions and give characterization of geometric properties, type p, of Banach spaces. In the sequel, let B be a real separable Banach space with norm

II . II.

Write {X n } ~ X if there exists a constant C > 0 such that, for sufficiently large z , n, " L P(IIX;II > x) ::::; CnP(IX! > e), where X is a real random variable.

;=1

Write {X n }

L" P(IIX;II > x)

;=1

1 Received

~

X

if there exists a constant C

~ CnP(IXI

> z},

>

0 such that, for sufficiently large x,n,

where X is a real random variable.

July 13,1999. Supported by the Science Fund of Tongji University

230

ACTA MATHEMATICA SCIENTIA

Vo1.21 Ser.B

A Banach space B is called type p (1 :'S p:'S 2) if for any zero mean B-valued independent

random element sequence {X n, n 2: I}, there exists C n

n

i=l

i=1

= C p > 0 such

that

Our main results are as follows: Theorem 1 Let 1 :'S t < 2. Suppose that 1( x) > 0 is a slowly varying function as x The following statements are equivalent:

-+

+00.

(a) B is of type p for some t < p:'S 2; (b) Suppose r > 1. For each sequence {X n } of zero mean B-valued independent random elements, if (1.1 ) then \I€

> 0 we have

L n r00

21(n)P(IISnll

2:

€.

n 1/t ) <

n=1

Ln n=1 00

r

-

21(n)P(

max IISkli > e- n

1~k~n-

+00,

1/t ) <

(1.2)

+00,

(1.3)

00

(1.4) (c) Suppose that l(x) is non-decreasing. For each sequence {X n } of zero mean B-valued independent random elements, if (1.5) then \I€

> 0 we

have (1.6)

(1.7) (d) Suppose that 1(x) satisfies, for every positive integer k k

L 1(2

i

)

:'S C· k .1(2k ) .

(1.8)

i=1

For each sequence {X n } of zero mean B-valued independent random elements, if (1.9) then \I€

> 0 we

have (1.10)

Theorem 2 'Let 0 < t < 2, l(x) > 0 be a slowly varying function as x {Xn } be a sequence of zero mean B-valued independent random elements.

-+

+00,

and let

Liang & Wang: CONVERGENCE RATES IN THE LAW OF LARGE NUMBERS

No.2

231

(a) Suppose r > 1 and {Xn} ~ X. If one of the sums holds in (1.2), (1.3) and (1.4), then E[IXlrt1(IXnJ < +00. (b) Suppose {Xn} ~ X and l(x) is a non-decreasing. If one of the sums holds in (1.6) and (1.7), then E[IXlt1(IXnJ < +00. (c) Suppose that {Xn} is iid r.v.'s and that l(x) satisfies positive integer k and l(n)logn

2:

s > 0 for n 2: no > O.

k

L

i=1

1(2 i )

2: C· k .1(2k ) for every

Then (1.10) implies

Theorem 3 Let 0 < t < 1, l(x) > 0 be a slowly varying function as x {X n } be any sequence of B-valued random elements.

->

+00, and let

(a) Suppose r > 1. If {Xn} is independent and (1.1) is satisfied, then (1.2), (1.3) and (1.4) are equivalent and hold. (b) Suppose l(x) is non-decreasing. If (1.5) is satisfied, then (1.6) and (1.7) are equivalent and hold. (c) Suppose that l(x) satisfies (1.8). Then (1.9) implies (1.10). Let {Xnd be an array of rowwise independent real-valued r.v.'s with EX nk = 0 and P(!Xnk ! > x) ::; P(IXI > x) for all n, k and x > O. If EIXl u < 00 for some 1 ::; t < 2, then Corollary 1( Hu, M6ricz and Taylor (1989), Theorem 2)

1 n o o n 1t n 1/t LXnk -> 0 completely, i.e. LP(IILXnkli 2: m / ) < 00, "IE> O. k=1 n=1 k=1 In this section, let {Tn' n 2: I} be a sequence of non-negative, integer valued random variables and T a positive random variable. All random variables are defined on the same probability space. x

->

Theorem 4 Let 1 ::; t < 2. Suppose that 1( x) > 0 is a slowly varyiny function as +00. The following statements are equivalent: (a) B is of type p for some t < p::; 2;

(b) Under the assumptions of Theorem l(b) and (d), respectively, if there exists "/ > 0 00 such that L n r- 21(n)P( ~ < ,,/) < +00. Then "IE > 0, .. =1

00

L

n r- 21(n)P(IISTnil 2: E' T;/t) < +00 .

.. =1

(c) Under the assumptions of Theorem l(b)-(d), respectively, if there exsists EO that

00

L

.. =1

nr-21(n)P(I~

-

TI

> EO) < +00, with P(T::; B) 00

L

= 1 for some B

> 0 such

> 0 , then "IE > 0,

n r- 2Z(n)P(IISTnil 2: E' n 1/t) < +00 .

.. =1

Let r > Land let {X..} be a sequence of iid real valued r.v.'s. Assume that 00 +00, and for any given E > 0, L nT-2P(I~ - TI > E) < +00, where P(A ::; T ::;

Remark

r < EIX 11

.. =1

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ACTA MATHEMATIC A SCIENTIA

Vo1.21 Ser.B

= 1 for

some real numbers A, B , such that 0 < A < B < +00, Lin (1986) proved that Tn L: n T- 2P(1 L: (Xk - EXk)1 ~ f · Tn) < +00. n=1 k=1 It is easy to show that if P(A :S T :S B) = 1 for some real number A, B, where 0 < A <

B) 00

< +00 or P(A :S T) = 1 for some A> 0, P(I~ - TI > f). So, the above result of Lin B

then for any given e > O(f < A), P( :; < A - f) :S is a special case of our results in real space. In the sequel, C denotes a positive constant whose value may change at different place.

2

Proofs of Main Results It is well-known that if l(x)

> 0 is a slowly varying function as z

-+

+00, then

1 \-It 0 rImk--++oo SUP2"~x~2"+1 . ..!iEl. 1 10 uIm x--++ oo ~ I(x) = , v >; 1(2") = . 61(x) 0 61(x) 3 lim x--++ oo x +00, lim x--++ oo x0, Vb> O. Lemma 1 (Shao (1988), Corollary 2.4) Let {X n } be a sequence of zero mean independent random elements in a type p space B. Then for q ~ p, there exists a positive constant C = Cp n such that Emaxl
=

Firstly, we prove (1.3). Let Yij

=

= XjI(IIXjll

:S 2i/ t), Sik

by the properties of l(x) we have 00

' " n T- 21(n)P( max L...J l
k

= L: Yij, j=1

T;k

= Sk -

Sik, then

00

IISkl1 ~ f · n 1/t) :S C'" 2i(r-l)1(2i)P( max IITikl1 ~ :.. 2i/ l ) L...J l
-

00

+C'" 2i(r-l)1(2 i)P( max IISikll ~ :. . 2i/t) L...J l
+C

It + lz + C.

=:

00

Note that, from Lemma 1 of Bai and Su (1985) we have E[lXjTtl(IXji)J

L: 2iTl(2i)P(IXI > 2i/ l ) < 00

i=O


2 i +1

i=O

j=1

¢:::=}

< +00, which follows II :S C L: 2i(r-l)1(2 i) L: P(IIXjll > 2i / t ) < +00. For

t < s < rt, by the property 30 of l(x), the condition (1.1) implies EIXls < +00, hence, from EX n

= 0 we have

(2.1) Thus, to prove I 2 < 00

Ii

00, it is sufficient to show that

2i(r-l)1(2i)P(maxl~k<2i+'IISik - ESik11 ~ f· 2i/t) < +00,

=: L

"If> O.

i=O

In fact, choosing q

> max{p, PI~r_~I), rt}, 00

Ii :S

and using Lemma 1 we have

2 i +1

2 i+1

CL 2i(r-l- q/I)1(2i){[L EIIYijIlPJq/p + L EIIYijllq} =: i=O j=1 j=1

t, + I 4 ·

If ri :::; p,

L 00

13

:::;

233

Liang & Wang: CONVERGENCE RATES IN THE LAW OF LARGE NUMBERS

No.2

C

2i(r-l- q/t)l(2 i

)[L

i=O

j=l

:::; C

t;

:::; C

+C L

00

1

2;+1 , 2 i

0

2i(r-l-q/t+q/Pll(2i) + C 00

2i (r -

l - !f + t

ll(2 i)[

x p/ t- l P(IIXjW

t;

~

> x)dx]q/p i

2i(r-l-q/t+q/P)l(2i )[k~o

L 2krl(2 k)P(IXl t > 2 i

k- l

1,-1 2k

x p/ t- l P(IX It

> x)dx]q/p

)]t (1 + 6).2f (l(2 i))- t

k=ko

i=O

< C + C L: 2i(r-l-rq/p+q/p)(l(2i))l-q/P(1 + 6)qi/ p < +00, 00

i=O where 6 > 0 is small enough. Similarly, if rt > p,

13

:::;

C

L: 2i(r-l-q/t+q/p)l(2i) 00

i=O 00

+CL: 2i(r-l-q/t+q/Pl(l(2i))l-q/P(1 + 6)qi/ p{E[lXl rtl(IXlt)]P/p < +00. i=O C ~ 2i(r- q/t)l(2 i) + C ~ 2krl(2 k)P(IXl t > 2k - l ) < +00. i=O k=l Next, we prove (1.4). By the property of lex) and Lemma 1 of Bai and Su (1985) we have

As in the above proof we get 14

00

00

:::;

00

00

00

Similarly to the proof of (1.3), (1.4) is proved. (a)=}(c) Clearly, we only need to prove (1.7). Let the definition of (a)=}(b). By the properties of lex) we have

Yij, Sih T ik

be as in

~ len) P( max II Sk'll2: 1'"' nl/t) < C~ l(2 i)P( max IITikll2: ~. 2i/ t) L.J n l
n=l

-

00

i=O-

-

+C'" 1(2 i)P( max L.J l
IISik II 2: -21'" . 2i/ t) + C

=:

Is + 16 + C.

From E[IXltlCIXji)] < +00, it is easy to prove Is < +00. Since lex) is a positive non-decreasing function, the condition (1.5) implies EIXl t < By EX n

= 0 we get

00

IIESikll/2i/t:::; C2 i P(!Xjt > 2i) + C'" 2kP(iXl t > 2k) --+ O.

max

L.J k=i

l
-

Thus, to prove h <

00, it suffices to show that

L: l(2i)P(maxl~k<2;+11ISik 00

1~ =:

i=O

ESik11

2: I'" . 2i/ t) < +00,

VI'"

> O.

+00.

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ACTA MATHEMATIC A SCIENTIA

Vo1.21 Ser.B

In fact, from the definition of type p and similarly to the proof of 14 we get 00

1~:S "L1(2i)2i(1-P/t) +CE[lXltl(IXn]

< +00.

i=O

(a)=>(d) By the properties of l(x) and (1.8), similarly to the proof of (a)=>(c) we get

(b)=>(a), (c)=>(a) and (d)=>(a) We only prove (b)=>(a), the proof of(c)=>(a) and (d)=>(a) is analogous. Suppose that (b) is satisfied. In (b), take l(x) = 1, and further let {X n } be independent and symmetric. Note that (1.3) and (1.4) imply (1.2). Thus, if EIXlrt < +00, 00

then we have 2:: n r - 2 P(IISnll ?: E' n 1 / t ) n=l

< +00,

Sn/n 1/t

--t

'rjE > O. which implies

(2.2)

0 a.s ..

By (2.2), similarly to the proof of Theorem 1 in Liang and Niu (1998), we can verify that B is of type p for t < p :S 2. Proof of T'heor-em 2 (a) It suffices to show that (1.2)=> E[IXlrtl(iXl t)] < +00. Define

X~ = X n - X~, where X~ denotes independent copy of X n and set S~ is a sequence of independent symmetric B-valued r.v.' s, and

"L nr-21(n)P(IIS~11 00

?: E' n 1/t) < +00,

= 2::7=1 X],

'rjE> 0,

Then {X~}

(2.3)

n=l

which implies from r > 1 that (2.4) By using (2.3) and (2.4) we can get n

"L P(IIX;;II ?: 2n 1/t) :S 4P(IIS~11

k=l

?: n

1/t), n?: n1

(2.5)

and choose n ?: nz such that P(IIS:'II ?: n 1/t) < 8.~j' Thus, by Proposition 6.8 of Ledoux and Talagrand (1991) we get EIIS:,/n1/tW < 00. While using EXn = 0 we get EIISn/n1/tll :S EIIS:'/n1/tll. Therefore, we ha~e (2.6) By (2.6), and using the Ottaviani inequality (cf: Wu and Wang (1990), P15), we get P(lIXkll ?: n 1 / t) :S P(max1~k~n IISkl1 ?: !n 1/t) :S CP(IISnll ?: ~nl/t) < for n sufficiently large, and hence by symmetrization inequality (cf: Wu and Wang (1990), P.114) and (2.5)

!

nP(IXI?: n1/t):s C'tP(IIXkll?: n1/t):s C'tP(IIX;;II?: k=l k=l 00

which follows from (2.3) that 2:: nr-1l(n)P(IXI ?: n 1/t) n=l Su (1985), we get E[lXlrtl(IXl t)] < +00.

~n1/t):S CP(lIS~II?: ~n1/t)

< +00, hence, by Lemma 1 of Bai and

Liang & Wang: CONVERGENCE RATES IN THE LAW OF LARGE NUMBERS

No.2

We can similarly prove (b). Next, we prove (c). Note that for every j

U U {IISk -

i=j k=2 i

S2;11 ~ 2E' 2

il t

~

235

1,

U U {IISkll ~ E' 2i l t } ,

} C

i=j-l k=2 i

then from Borel-Cantelli lemma and noticing that {max2i:Sk<2i+l IIsk2~/~,dl, i ~ I} is indepen> . ) = 1 an d d ent, we h ave P( max2i:Sk<2i+l IIsk-s,dl 2;/' _ 2E, 2.0. P(sup max IISk ~ S2 . 1 2'It i?j 2 i :Sk<2·+

i

II > 2E) = 1 ,

Vj ~ 1.

(2.8)

While, from (2.8)

00

= CI:l(2j ) = 00, j=1

which is in contradiction with the assumption. ,",00 P( max2i:Sk<2i+l II s k2-i /s,dl S0, 6i=1 ~ 2) E < 00, . P(sup . max.

.> . 2'
'_J

IISk - S2i 'I

2'

t

II

W hiICh

iImp l'ies

I: oo

1 'It ~ 2E) ~ P( max .IISkll ~ 2E' 2' ). 2"'=J l
Therefore

The proof of the rest is similar to that in (b).

Proof of Theorem 3 (a) By the property 3° of l(x), we know that the condition E[lXlrtl(lXl t)] < +00 implies EIXl t < +00, it is easy to verify

Sn/n 1/ t ~ O.

(2.9)

By (2.9) and using the Ottaviani inequality we have (2.10)

for n sufficiently large. "IE > 0, when 2 m Ottaviani inequality we can obtain that

< n :::; 2m +1 for m sufficiently large, using the (2.11)

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ACTA MATHEMATICA SCIENTIA

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By the property of l(x), and using Lemma 1 of Bai and Su (1985) and (2.11) we have (2.12) Hence, by (2.10) and (2.12)) we obtain that (1.2), (1.3) and (1.4) are equivalent. Thus, we only need to prove that (1.2) holds. By symmetrization inequality, we may suppose that {X1\} is 1\ symmetric. Let Y"i = X;I(IIXill n 1 / t), S:, = L: Y1\i, S:: = S" - S:" obviously i=1

:s

00

"" n r- 21(n)P(IIS1\1I 2: e- n 1 / t) L...J 1\=1

00

:s C"" 2i(r-1)1(2i) . max: P(IIS~II 2: ~ . n L...J 2'<1\<2'+' 2

1

/

t)

i=O-

00

+c2.=2 i(r-l)I(2i) . max P(IIS:,II2: ~. n 1 / t) + C i=O 2·~1\<2·+1 2

•

=: J 1

+ h + C.

Similarly to the proof of 11 we can get h < 00 and S~/nl/t !:... 0, further we obtain S~/nl/t !:... 0 from (2.9), and using Lemma 3.1 of De Acosta (1981) we have EIIS:,/n 1 / t ll ---> 0, as n --+ +00. Thus, to prove J 2 < 00, it suffices to show that

J; =: 2.= 2i(r-l)I(2i)max2;~1\<2;+IP(IIIS~II00

EIIS:,III 2: f · n 1/ t ) < +00,

'iff> O.

i=O

In fact, if q > max{2, 2t~r~ 11, rt}, using Theorem 2.1 of De Acosta (1981) and as in the proof of I; we get

J; :s C L 00

,=0

2

i

(r -

1ll(2 i ) 2;~;;~;)n-q/t EIIIS:.II- Ells~lllq] <

-

00.

The proof of (b) and (c) is similar to that (c) and (d) in Theorem 1, respectively, so omitted here.

IS

Acknowledgement I would like to express my thanks to Prof. Hu Dihe for his encouragement and valuable help. References 1 Bai Z D. Su C. The complete convergence for partial sums of i.i.d.random variables. Sci Sinica (Ser A), 1985, 28: 1261-1277 2 Baum L E, Katz M. Convergence rates in the law of large numbers. Trans Amer Soc, 1965, 120(1): 108-123 3 De Acosta A. Inequalities of B-valued random vectors with applications to the strong law of large numbers. Ann Probab, 1981, 9: 157-161. 4 Hu Y C, M6ricz F, Taylor R L. Strong laws of large numbers for arrays of row wise independent random variables. Acta Math Hung, 1989,54: 153-162 5 Ledoux M, Talagrand M. Probablity in Banach Spaces. Berlin: Springer, 1991 6 Liang H Y, Niu S L. p-type space and convergence rates of tail probabilities for sums of B-valued random elements. Acta Math Appl Sinica, 1998.21(1): 92-102 7 Lin Z Y. Complete convergence for randomly indexed stuns. Chinses Ann Math, 1986,7A:298-302 8 Shao Q M. A moment inequality and its applications. Acta Math Sinica, 1988, 31(6): 736-747 9 Wu Z Q, Wang X C. Probabilty in Banach Space (In Chinese). Changchun: Jilin University Press, 1990