A weak law for normed weighted sums of random elements in rademacher type p banach spaces

1 } is stochastically dominated by a random element V in the sense that for some constant D < co,

t>O, n>l.

Pi IIVnll’ t> G DP( IlDf’ll > t>,

(2.1)

Let {a,, n&l} and {b,, n > 1 } be constants with a,, # 0, b, > 0, n 3 1, and supposethat either

$1,

i

”

and

lajlp=4b:),

j=l

i j=l j’ bj” lajl’

or

icl lajlp= O(n Ia,Ip),

and

f: j=l j2 bj” Iail’ =O ( z~=b”lajlp) J 1

(2.3)

and

,g, lajlp= O(n la,lP)

(2.4)

or

p>L--bn t, n IanI or

b"f n IaA

and

i

j=l

Ia,-I=0

V-5)

hold. Zj-

nP

(2.6)

then the WLLN

cy=l aj(vj-EVjZ(IIVjII b, obtains.

%llanl))L

0

(2.7)

262

ADLER,

ROSALSKY,

AND

TAYLOR

Let

Proof:

Note at the outset that E // Vn,il < cc, 1 < j 0,

p IICy= 1 aj( vj- vnj)ll b, 6 i

p( IIvjll > cn)

j=l

c,> =0(l)

(by (2.1))

(by (2.6 1);

whence

Thus, it suffices to show that

CT= 1 aj ( Vnj- EJ’nj) P, 0 b,

(2.8)

To this end, for arbitrary E> 0,

P


E n

i

IIj=l

aj(Vnj-EV,,j)

c n

I(

de j?l lajl’ E IIVnj-EVnjllP

(since 55’ is of Rademacher type p)

6 5 ,$ 1aj-I’E IIVnjll” PI

employing

/=I

integration

by parts (see Lemma 1 of Adler and Rosalsky [3]).

A WEAK

The proof will be completed

LAW

OF LARGE

263

NUMBERS

once it is verified that (2.9)

j=l

and

jc, l”jlPE IIvlI”z(llovll

(2.10)

GCn)=o(bI)

hold. Note that under (2.5), p may and will be taken to be 1 without any loss of generality. To prove (2.9), observe that under (2.2),

~ Ccy{ IIDVII > 4 =CnP(llmq n(c;/n’)

>c,}=o(l)

by (2.6). On the other hand, under (2.3), (2.4), or (2.5)

again by (2.6) and so (2.9) obtains. To prove (2.10), note that c,T under (2.2), (2.3), (2.4), or (2.5) and that (2.3), (2.4), and (2.5) individually ensure i

(2.11)

lujlp = o(b;).

j=l

Thus, d,=c,/n,

(2.11) holds under (2.2), (2.3), (2.4), or (2.5). Let co=0 n>l. Define an array (Bnk, O
!1

l~~l~)(~“~-~‘)

0

for

l
for

k=O,

and

n>2

n, n>l.

It will now be shown that (Bnk, 0 < k < n, n > 1 } is a Toeplitz array; that is,

f k=O

I&

= O(1)

(2.12)

264

ADLER,

ROSALSKY,

AND

TAYLOR

and B nk

+

as n -+ a for all fixed k 3 0.

o

(2.13)

Clearly (2.11) entails (2.13). To verify (2.12), note that B,,>O, Odk
Thus, (k+

implying

for k Z 1 that

CPk+l-C::

Jk+l)Pd;+l-kPd: k

k

1)P-kkP<3kP-1,

k> 1,

<(kP--+3k~-2)d~+1-kP--d~.

(2.14)

Then under (2.2), since d,J, cpk+ 1 -c; k

3d; G-c-k2-P

3~; k*’

kZ 1;

whence for n > 2, i

B -(b; < 3

f: I4 .IP)(;;;

j=,

$)=O(l)

k=O

and so (2.12) holds. Now under (2.3), (2.4), or (2.5), for 1~22, ioBnkG(k

jl

lajl’)(~~~

((k’~1+3kP~2)d~+~-kY~‘d~))

(by (2.14))

(2.15)

A WEAK

LAW

OF LARGE

NUMBERS

265

Under (2.3), for n 2 2,

Under (2.4) or (2.5), for n > 2,

4t) G? (!, lajlp)(~~~ kpp2) (since =

i

-& O(np-‘)

if

(2.4) holds

C O(log n) log n

if

(2.5) holds

= O(1). Thus, under (2.3), (2.4), or (2.5), recalling (2.15),

i 42,=0(l) k=O

and again (2.12) holds, thereby proving that { Bnk, 0 < k $ n, n > 1 } is a Toeplitz array. By (2.6) and the Toeplitz lemma (see, e.g., Knopp [ 13, p. 743 or Lo&e w, P. 2501)~ i

&M(IIWl

>c,c) =0(l).

k=O

Next. note that

6 jgI lailp E IIVIP W)vII =d

,i lajl’ i n ,=I

k=l

64

E IIvllpZ(ck- I < IID~II G ck)

(2.16)

ADLER, ROSALSKY, AND TAYLOR

266

>Ck) + c (G+,-c~)~(llDv/l k=l

(by the Abel “summation
jcl

luj,p

nc’

of parts” lemma) c~+~-cSkP{~~~~~l>ck}+o(l)

(by (2.11))

k=l

=D-p

i

B,,&P{IIDVjl

>c,}+o(l)

k=O

=0(l)

(by G’.16)),

thereby establishing

(2.10) and the theorem.

1

3. FINAL REMARKS

Some remarks pertaining (i)

to the theorem are in order.

If the hypotheses of the theorem obtain with p = 1, then

IIC ujEvjz( II vjll GCn) G C IQjI E IIvjII Z(IIvjII GCn)=o(bn) j=l

II

j=l

was shown in the proof of (2.8) and, consequently,

Moreover, a perusal of the argument reveals that the hypothesis of independence is not needed when p = 1. (ii) The example of weighted i.i.d. random variables presented in Section 5 of Adler and Rosalsky [4] satisfies the hypotheses of the theorem and hence the WLLN (2.7) obtains. On the other hand, Adler and Rosalsky [4] showed that the corresponding SLLN fails. It should be noted that the random variables in that example are not integrable and the real line is of Rademacher type p for all 1

1, and ( V,, n 2 1 } a sequence of symmetric, i.i.d. random elements in a real separable, Rademacher type p (1

A WEAK LAWOF

267

LARGENUMBERS

(iv) The following example (due to Beck [7]) theorem can fail even for well-behaved sequences {b,, n> I} when

shows that the {a,, ~12 1 } and

(3.1) Consider the real separable (Rademacher type p = 1) Banach space I1 of absolutely summable real sequences v = {vi, i2 1 > with norm IlUll =CIFl [Vii. Let V (n) be the element having 1 in its nth position and 0 elsewhere. Define a sequence { V,, n 2 1 } of random elements in I1 by requiring the ( V,, n 2 1 } to be independent with p{v,=v’“‘)=p(v,=

-v(n)}=f,

n> 1.

Let a, = 1, b, = n, n 2 1. Then (2.2), (2.3), (2.4), and (2.5) all fail but (3.1) holds. Also, (2.1) (with V= I/, and D = 1) and (2.6) both hold. Since

II III jg, vi

n = 1 a.c.,

the conclusion (2.7) of the theorem fails. It should be noted that I, is not of Rademacher type p for any 1

1 } considered above that ’ = np,

i

EIIVjllP=n,

nal.

j=l

ACKNOWLEDGMENTS The authors are grateful to the referee and associate editor for their helpful comments and constructive criticism which led to substantial improvements in the paper. The referee was also helpful in supplying some additional references which are relevant to the types of problems that are of interest to the authors.

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ADLER,

268

ROSALSKY,

AND

TAYLOR

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