Download ON THE STRONG LAW OF LARGE NUMBERS FOR SEQUENCES

PROBABILITY
AND
MATHEMATICAL
STATISTICS
Vol. 27, Fasc, 2 (2007), pp. 205-222
ON THE STRONG LAW OF LARGE NUMBERS FOR SEQUENCES
OF BLOCKWISE INDEPENDENT
AND BLOCKWISE p-ORTHOGONAL
RANDOM ELEMENTS IN RADEMACHER TYPE p BANACH SPACES
BY
ANDREW ROSALSKY
AND LE VAN THANH
(GAINESVILLE,FLORIDA)
(VINH CITY, VIETNAM)
Abstract. For a sequence of random elements {~, n ~ 1} taking
values in a real separable Rademacher type p (1 ~ p ~ 2) Banach space
and positive constants b; i 00, conditions are provided for the strong
law of large numbers
2:;=1 Vi/b" ---+ 0 almost
surely. We treat the fol-
lowing cases: (i) {V,,, n ~ 1} is blockwise independent with EV" = 0,
n ~ 1, and (ii) {~, n ~ 1} is blockwise p-orthogonal. The conditions
for case (i) are shown to provide an exact characterization of Rademacher type p Banach spaces. The current work extends results of M6ricz
[12], M6ricz et al. [13], and Gaposhkin [8]. Special cases of the main
results are presented as corollaries and illustrative examples or counterexamples are provided.
2000 AMS Mathematics
Secondary: 60Bll, 60B12.
Subject Classification: Primary 60F15;
Key words and phrases: Blockwise independent random elements, blockwise p-orthogona1 random elements, strong law of large
numbers, almost sure convergence, Rademacher type p Banach space.
1. INTRODUCTION
M6ricz [12] introduced the concept of blockwise independence for a sequence of (real-valued) random variables and extended a classical strong law of
large numbers (SLLN) of Kolmogorov (see, e.g., Chow and Teicher [6], p. 124)
to the blockwise independent case. (Technical definitions needed in this paper
will be discussed in Section 2.) Gaposhkin [7] and [8] also studied the SLLN
problem for sequences of blockwise independent random variables; in those
papers he also proved SLLNs for sequences of blockwise orthogonal random
variables.
In the present paper, we consider a sequence of random elements
{~, n ~ 1} defined on a probability space (Q, !F, P) and taking values in a real
206
A. Rosalsky
separable
SLLN:
Banach space
lim
n.....oo
f![
and Le Van Thanh
with norm 11'11. We provide conditions
I~-i
Vi
l"b
=0
for the
almost surely (a.s.),
n
where {bn, n ~ 1} is a sequence of positive constants with b; i 00. The Banach
space f![ is assumed to be of Rademacher type p (1 ~ p ~ 2). The main findings
are Theorems 3.1, 3.2, and 3.3. In Theorem 3.1 the random elements {li;" n ~ 1}
are assumed to be blockwise independent with Eli;, = 0, n ~ 1, whereas in
Theorem 3.3 the random elements are assumed to be blockwise p-orthogonal. In
Theorem 3.2, it is shown that the implication (3.2) ~ (3.3) in Theorem 3.1
indeed provides an exact characterization
of Rademacher type p Banach
spaces. Theorems 3.1 and 3.3 are very general results in that they are new when
the Banach space f![ is the real line R. Of course, special cases of Theorems 3.1
and 3.3 are known to hold when f![ = R.
The present work extends results of M6ricz [12], M6ricz et al. [13], and
Gaposhkin [8]. Our proofs are substantially different from those of the earlier
counterparts due to a recent and elementary result of Chobanyan et al. [5]
(Lemma 2.5 below) and these differences will be discussed in Remark 3.2.
The plan of the paper is as follows. Technical definitions, notation, lemmas
and other results used in the proofs of the main results or their corollaries are
given in Section 2. The main results are stated and proved in Section 3. In
Section 4, some corollaries and interesting examples or counterexamples are
presented.
2. PRELIMINARIES
Some definitions, notation, and preliminary results will be presented prior
to establishing the main results. Let f![ be a real separable Banach space with
norm 11'11. A random element in f![ will be denoted by V or li;" etc.
The expected value or mean of a random element
denoted by EV: is
defined to be the Pettis integral provided it exists. That is, V has an expected
value EV Ef![ if f(EV) = E(f(V)) for every f Ef![*, where f![* denotes the (dua0
space of all continuous linear functionals on f![. If E IIVII < 00, then (see, e.g.,
Taylor [18], p. 40) V has an expected value. But the expected value can exist
when E IIVII = 00. For an example, see Taylor [18], p. 41.
Let {~, n ~ 1} be a symmetric Bernoulli sequence; that is, {~, n ~ 1}
is a sequence of independent and identically distributed (i.i.d.) random variables with P {Yi = 1} = P {Yi = -1} = 1/2. Let f![oo = f![ x f![ x f![ x .. , and define
v:
00
~ (f![)
= {(Vi, V2, ... ) E f![00:
I~
Vn
n= 1
converges in probability}.
207
Strong law of large numbers in Banach spaces
Let 1 ~ p ~ 2. Then q; is said to be of Rademacher
a constant 0 < C < 00 such that
OC!
E
II I
type p if there exists
OC!
P
Y;, vnll
C
~
n=1
I Ilvnll
P
for all
(V1'
... ) E ~
V2,
(q;).
n=1
Hoffmann-Jorgensen and Pisier [9] proved for 1 ~ p ~ 2 that a real separable
Banach space is of Rademacher type p if and only if there exists a constant
o < C < 00 such that
n
(2.1)
Ell
n
I ViII
P
I EIIViW
C
~
i=1
i=1
for every finite collection {V1, ••• , ~} of independent mean 0 random elements.
If a real separable Banach space is of Rademacher type p for some
1 < p ~ 2, then it is of Rademacher type q for all 1 ~ q < p. Every real separable Banach space is of Rademacher type (at least) 1 while the .Pp-spaces and
lp-spaces are of Rademacher type 2 /\ P for p ;?: 1. Every real separable Hilbert
space and real separable finite-dimensional Banach space is of Rademacher
type 2. In particular, the real line R is of Rademacher type 2.
A finite collection of random elements {V1, ... , VN} (N ;?: 2) is said to be
p-orthogonal (1 ~ p < (0) if E II~IIP < 00 for all 1 ~ n ~ Nand
m
n
E
II I
i=1
an(i)
~(i)IIP
~
E
II I
P
an(i)
Vn(i)II
i=1
for all choices of 1 ~ n < m ~ N, for all constants {af, ... , am}, and for all
permutations TC of the integers {1, ... , m}. A sequence of random elements
{~, n > 1} is said to be p-orthogonal (1 ~ p < (0) if {V1, ... , VN} is p-orthogonal for all N ;?: 2. The notion of p-orthogonality was introduced by Howell and
Taylor [10]; we refer to Howell and Taylor [10] and M6ricz et al. [13] for
a detailed discussion of p-orthogonality.
Let {13k, k ;?: 1} be a strictly increasing sequence of positive integers with
131 = 1 and set Bk = [13k, 13k + 1), k;?: 1. A sequence of random elements
{~, n ;?: 1} is said to be blockwise independent (resp., blockwise p-orthogonal
(1 ~ p < (0)) with respect to the blocks {Bk, k ;?: 1} if for each k ;?: 1 the random elements {Vi, i E Bk} are independent (resp., p-orthogonal). Thus the random elements with indices in each block are independent (resp., p-orthogonal)
but there are no independence (resp., p-orthogonality) requirements between
the random elements with indices in different blocks; even repetitions are permitted.
The following notation will be used throughout this paper. For x ;?: 0, let
Lx J denote the greatest integer less than or equal to x and let Ix l denote
the smallest integer greater than or equal to x. We use Log to denote the
logarithm to base 2. The symbol C denotes a generic constant (0 < C < (0)
which is not necessarily the same in each appearance.
208
A. Rosalsky
For {13k, k ;::: 1} and {Bb
notation:
Bkm) = B
and Le Van Thanh
k ;::: 1} as above, we introduce
n [Z", 2m + 1),
k
k ;:::1, m ;:::0,
= {k;::: 1: Bkm) # 0},
L;
rkm) = min Bkm) ,
m;::: 0,
m
kEIm,
= card 1m,
Cm
the following
> 0,
m;::: 0,
00
q>(n)
I
=
Cm I[2m,2m+
(n),
1)
n ;:::1,
m=O
denotes the indicator function of the set [Z", 2m+i), m
It is easy to verify that the following relations are satisfied:
(i) If 13k = 2k- \ k;::: 1, then
where
I[2m,2m+1)
(2.2)
Cm
(ii) If 13k
= L
= 1, m > 0,
«:
(2.3)
J
Cm
(iii) If 13k
(2.4)
(iv) If 13k
(2.5)
(1)
= (!)
L 2k/(Logk)C< J
Cm
= (D((Logm)a)
and
q>(n)
and
= (D(m(i-a)/a)
Lka J ,
=
Cm
=
= 1, n > 1.
k
=
(!)
(1).
for all large k, where ex > 0, then
q>(n)
= (D((Log Log nY).
= L 2kC<J for all large k, where
(2.6)
(vi) If 13k
q>(n)
for all large k, where q > 1, then
=
Cm
(v) If 13k
and
and
q>(n)
°<
ex < 1, then
= (D((Log n)(i-a)/a).
> 1, where ex> 1, then
= (D(2m/a)
and
q>(n)
= (D(ni/a).
k, k;::: 1, then
(2.7)
Cm =
2m, m
> 0,
and
q>(n) ~ n, n
The following result is well known when
>
1.
= Rand
f!{
j
(2.8)
E(( max II I Villt) ~ C
i~j~n
where
p
= 2.
°
2.1. Let {~, n ;:::1} be a sequence of independent
mean
random
in a real separable Rademacher
type p (1 ~ p ~ 2) Banach space. Then
LEMMA
elements
> 0.
the constant
i=i
C does not depend
n
I EIIViII
i=i
P
,
n > 1,
on n.
Proof. Let!F" = o-(Xb 1 ~ i ~ n), n ;::: 1. Now it is well known but seems
to have been first observed by Scalora [17] that
1 Viii, !F", n ;::: 1} is a real
submartingale. In the case 1 < p ~ 2, by Doob's submartingale maximal
{III;=
209
Strong law of large numbers in Banach spaces
inequality
(see, e.g., Chow and Teicher [6J, p. 255), for n ~ 1 we have
n
~ c I E IIViII
P
(by (2.1)),
i=i
establishing
(2.8). In the case p
= 1, note that for n ~ 1
j
E( max
i~j~n
j
IIi=iI ViiI) ~
again establishing
(2.8).
E( max
i~j~ni=i
n
n
I IIViiI) = E( i=iI IIViIl) = i=iI
E IIViIl,
II
PROPOSITION 2.1 (Hoffmann-Jorgensen
and Pisier [9]). Let 1 ~ p ~ 2 and
let PI be a real separable Banach space. Then the following two statements are
equivalent:
(i) PI is of Rademacher type p.
(ii) For every sequence {~, n ~ 1} of independent mean 0 random elements
in PI, the condition
implies
,,~
V:
L.J=l
J
n
-+
0 a.s.
LEMMA2.2 (Howell and Taylor [10J). If PI is a real separable Rademacher
type p (1 ~ p ~ 2) Banach space, then there exists a constant C < 00 such that
n
E
n
III ViII
P
I
C
~
i=i
E
IIVi liP ,
n ~ 1,
i=i
for all p-orthoqonal sequences of PI-valued random elements.
LEMMA2.3 (Moricz et al. [13J). Let {~, n ~ 1} be a p-orthogonal
(1 ~ p < 00) sequence of random elements in a real separable Banach space and
suppose that there exists a sequence of nonnegative numbers {un, n ~ 1} such
that
m
m
EIIIViII
P
i=n
~
I
Ui
for all m ~ n ~ 1.
i=n
Then
P m
j
E(( m~x
n~J~m
IIi=n
I Villt) ~
(Log(2(m-n+
1)))
I
Ui,
m ~ n ~ 1.
i=n
An immediate consequence of Lemmas 2.2 and 2.3 is that if {Vi, ... , VN} is
a p-orthogonal (1 ~ p ~ 2) collection of N ~ 2 random elements in a real
210
A. Rosalsky
separable Rademacher
type p Banach space, then
and Le Van Thanh
j
(2.9)
E (m~x
n~J~m
III
ViIIY
i=n
P
1)))
~ C(Log(2(m-n+
m
I E IIViII
P,
1~ n~
m ~
N,
i=n
where the constant C does not depend on n, m, or N.
The next lemma is most probably known but we are unable to locate
a reference. It follows immediately from HOlder's inequality writing ai = ai'l,
as was so kindly pointed out to us by the referee.
LEMMA2.4. If a, ~ 0, 1 ~ i ~ n, and p ~ 1, then
n
n
(I aiY ~
I
nP-1
i= 1
i=
af·
1
The following elementary result was recently obtained by Chobanyan et
al. [5] and it will playa key role in the proofs of Theorems 3.1 and 3.3. Lemma 2.5 is closely related to a classical result of Prokhorov [14] and [15] when
f!( = R, the {~, n ~ I} are independent
symmetric random variables, and
b; = n", n ~ 1, with IX > O.
LEMMA2.5 (Chobanyan et al. [5]). Let {~, n ~ I} be a sequence of random elements in a real separable Banach space f!{, let {bn, n ~ I} be a nondecreasing sequence of positive constants, and let {kn, n ~ O} be a sequence of
positive integers such that
and
(2.10)
Then
·
1rm
(2.11)
n--+oo
if
and only
(2.12)
"'~
.L..t=
b
1
ll;t
n
= 0 a.s.
if
. maxkn~k<kn+lIII:=kn
1im
b
b
n --+ 00
kn + 1 -
Vill_
- 0 a.s.
kn
Remark
2.1. (i) Note that the first inequality of (2.10) ensures that
{kn, n ~ O} is strictly increasing and limn--+oo b; = 00.
(ii) It follows that if (2.12) holds for some sequence of positive integers
{kn, n ~ O} satisfying (2.10), then (2.12) holds for every sequence of positive
integers {kn, n ~ O} satisfying (2.10). Thus, in order to prove the SLLN (2.11),
nothing is lost in working with a convenient sequence such as k; = 2n, n ~ O.
This remark was made by Chobanyan et al. [5].
Strong law of large numbers in Banach spaces
211
(iii) If the random elements {lI;" n ~ 1} are independent and symmetric,
then by the random element version of Levy's inequality (see, e.g., Araujo and
Gine [3], p. 102) and the Borel-Cantelli lemma, (2.12) is equivalent to the
structurally simpler and apparently weaker condition
"kn+
· II '--'i=kn
1tm
b
1
b
kn+ 1 -
n-r co
17;11
-1
l
-
-
°
a.s.
kn
3. THE MAIN RESULTS
With the preliminaries accounted for, the main results may now be established. When f£ = Rand b; == n, a version of Theorem 3.1 was obtained by
Gaposhkin [8] using a substantially more complicated argument. Gaposhkin's
[8] condition is slightly different from (3.2) specialized to f£ = Rand b; == n
and is at least as strong. Gaposhkin's [8] result is an extension of an earlier
result of M6ricz [12] which was apparently the first SLLN for a sequence of
blockwise independent random variables.
°
THEOREM 3.1. Let {lI;" n ~ 1} be a sequence of mean
random elements in
a real separable Rademacher type p (1 ~ p ~ 2) Banach space f£ and let
{bn, n ~ 1} be a nondecreasing sequence of positive constants such that
(3.1)
and
If {lI;" n ~
and if
1} is blockwise independent with respect to the blocks {Bk, k ~ 1}
(3.2)
then
(3.3)
lim
n-r co
Proof.
I~-l
Vi
lb = ° a.s.
n
Set
j
1k(m) = max II
jeB~m)
I Viii,
kEIm'
m ~ 0,
i = r~m)
and
Note that, for m ~ 0,
ET!
~ -bP1
m
2 + 1
C~-l
I
kelm
E(1k(m»)p
(by Lemma 2.4)
212
A. Rosalsky
~
~C~-l
2m+ 1
C
and Le Van Thanh
I I EIIViII
P
ua.;
(by Lemma 2.1)
ieBkm)
I:=o
It thus follows from (3.2) that
ETl: <
ty and the Borel-Cantelli lemma
lim Tm
00,
and so by the Markov inequali-
= 0 a.s.
m--+w
Now it follows from the first inequality
of (3.1) that
max2m~k<2m+lIII:=2m Viii ~ Cmax2m~k<2m+lIII:=2m
---------
b2m+l-b2m
The conclusion
Viii
b2m+l
(3.3) follows immediately
~ CTm
from Lemma 2.5.
~
0 a.s.
III
Remark
3.1. (i) The slower b; i 00, the stronger is the assumption (3.2),
but so is the conclusion (3.3).
(ii) Theorem 3.1 is an analogue of an SLLN of Adler et al. [2] obtained
for a sequence of independent random elements in a Rademacher type
p (1 ~ p ~ 2) Banach space. The Adler et al. [2] result, which extends the
implication (i) => (ii) in Proposition 2.1 to more general norming constants
o < b; i 00, is a random element analogue of a classical result of Kolmogorov
(see, e.g., Loeve [11], p. 250). It should be pointed out that this SLLN of Adler
et al. [2] does indeed follow immediately from Theorem V.7.S (or Corollary
V.7.S) of Woyczynski [19] and the Kronecker lemma.
(iii) When p = 1, Theorem 3.1 is not of interest since the mean 0 assumption, the blockwise independence assumption, and (3.1) are not needed. Indeed,
for a sequence of random elements {~, n ~ 1} in a real separable Banach space
and constants 0 < b; i 00, Cantrell and Rosalsky [4] recently proved that if
IE ( llViII+b
IIViii ) <
assumption than I: E Vill/b
00
i=l
(which is a weaker
i
1
II
i
< (0), then
n
lim
n--+w
I Vi/b
n
= 0 a.s.
i= 1
irrespective of the joint distributions of the {~, n ~
(iv) If (3.1) holds where 0 < b, i and
(3.4)
Cm
= o (am)
for all a>
1,
1}.
Strong
law of large numbers
then a sufficient condition
213
spaces
for (3.2) to hold is that
EllliiliP
00
(3.5)
in Banach
i~~<oo
for some O<q<p.
To see this, it follows from the first inequality of (3.1) that b2n+ 1/b2n
for some b > 0 and all n ~ O. Thus for all m ~ 1 we have
m-1b
(3.6)
= b,
b2m
TI -b-.
2j+1
j=O
~
1+b
m
~ bd1 +b) .
2J
Hence
m
~ m~o
i~m
~c I
00
m=O
=
c
2m+ 1-1 E
Ill/;IIP
I __
ql_
IE ";"b
(by (3.6))
(by (3.4) with a
= (1 + b)(P-q)/(p-l»)
bi
i=2m
P
i=l
thereby establishing
E IlliillP C~-l
b1 bf-q (1 + b)(p-q)m
1
2 + -l
00
<
00
(by (3.5)),
i
(3.2).
In the next theorem, we will show that the implication (3.2) => (3.3) holding
indeed completely characterizes Rademacher type p Banach spaces.
THEOREM 3.2. Let!!l' be a real separable Banach space and let 1 ~ p ~ 2.
Then the following two statements are equivalent:
(i) !!l' is of Rademacher type p.
(ii) For every sequence of mean 0 random elements {~, n ~ 1}, which is
blockwise independent with respect to some sequence of blocks {Bk, k ~ 1} and
every nondecreasing sequence of positive constants {bn, n ~ 1} satisfying (3.1),
the implication (3.2) => (3.3) holds.
Proof. The implication (i) => (ii) is precisely Theorem 3.1. To verify the
implication (ii) => (i), assume that (ii) holds. Let {~, n ~ 1} be a sequence of
independent mean 0 random elements in !!l' such that
(3.7)
214
A. Rosalsky
In view of Proposition
and Le Van Thanh
2.1, it suffices to verify that
I;=1 Vi ~
(3.8)
n
°
a.s.
Note that {~, n ~ 1} is blockwise independent with respect to the blocks
{[2k-1,
2k), k ~ 1}. Set b; = n, n ~ 1. Then (3.1) holds and, recalling (2.2), we
see that (3.7) and (3.2) are the same. Thus, by (ii), we see that (3.8) holds. l1li
b;
Bk
The next theorem, when specialized to {~, n ~ 1} being p-orthogonal and
n", n ~ 1, where a > 0, reduces to a result of M6ricz et al. [13] by taking
= [2k-1, 2k), k ~ 1, and recalling (2.2).
=
THEOREM 3.3. Let {~, n ~ 1} be a sequence of random elements in a real
separable Rademacher type p (1 ~ p ~ 2) Banach space and let {bn, n ~ 1} be
a nondecreasing sequence of positive constants satisfying (3.1). If {~, n ~ 1} is
blockwise p-orthoqonal with respect to the blocks {Bk, k ~ 1} and if
W (cp (i)r , < 00,
,~, E I~"P (Log
(3.9)
then
(3.10)
lim
,,"
LJ'-1
Vi
l"b
n->CXJ
=
"
°
a.s.
Proof. Define 1k(rn), kEIrn' m ~ 0, and Trn, m ~ 0, as in the proof of Theorem 3.1. Note that, for m ~ 0,
ETJ
~ -bP1
m
C~-1
2 + 1
C
~ ~c~-1
1
I
(Log(2cardBkrn»)t
C~-1
m
2 + 1-1
i~m
I
EIlViIIP
(by (2.9))
ieBkm)
2m+1-1
I
EIIViIlP
i=2m
1
~ C
(by Lemma 2.4)
C(Log2rn+1)P
~ _p-C~-1
b2m+l
b~m+1
E(1k(rn»)p
»a ;
2m+ 1
~
I
kelm
rn
C (Log 2 )P
2m+1-1
i~m
EIlViIIP
E 11T7;IIP
1
T(Logi)P(CfJ(i)t- .
I:=o
It thus follows from (3.9) that
ETJ < 00. The rest of the argument
exactly the same as that at the end of the proof of Theorem 3.1. l1li
is
Remark
3.2. We close this section with discussion of the main difference
between the structure of the proofs of Theorems 3.1 and 3.3 and that of the
215
Strong law of large numbers in Banach spaces
earlier results by Moricz [12], Moricz et al. [13], and Gaposhkin [8]. As in
Moricz [12] and Gaposhkin [8], let us now take b; = n, n ~ 1, and !!l' = R
(although Moricz et al. [13] took b; = n", n ~ 1, where (X > 0, and they assumed that the Banach space !!l' is of Rademacher type p (1 ~ p ~ 2». Moricz
[12], Moricz et al. [13], and Gaposhkin [8] bounded
1 Vi/n (where
2m ~ n < 2m + 1) by
I;=
(3.11)
I;=l Vii ~ II:~;l2m
I
Vii
n
1
+2m
max
2m<k<2m+1
Ii=2I Vii·
k
m
They then argued that each of the two terms on the right-hand side of (3.11)
converges to 0 a.s. as m ~ 00. Their arguments that the first term on the
right-hand side of (3.11) converges to 0 a.s. as m ~ 00 did not use any (type of)
independence hypothesis; only (a type of) orthogonality was used. However, in
our proof of Theorems 3.1 and 3.3, we only need to argue that the second term
on the right-hand side of (3.11) converges to 0 a.s. as m ~ 00. This is the case
because we then apply Lemma 2.5, and so we are in effect bounding
1 Vi/n
(where 2m ~ n < 2m+1) by
I;=
I;=1 Vii ~
f
I
r=O
max2r<k<2r+lII:=2r
2m
~
m
n
_
- 2m
r~o
Vii
Vii
2r max2r<k<2r+lII:=2r
2r
which via the Toeplitz lemma converges to 0 a.s. as m ~
and Theorem 9.1 of Chobanyan et al. [5]).
4. COROLLARIES
'
00
(see Proposition
3.1
AND EXAMPLES
In this section, some particular cases of Theorems 3.1 and 3.3 are presented as corollaries. Some illustrative examples or counterexamples are also
provided.
When the underlying Banach space is the real line, p = 2, and b; == n,
Corollaries 4.1, 4.3, 4.4 (take (X2 = 1), and 4.5 reduce to results of Gaposhkin [8]. This special case of Corollary 4.1 is also the SLLN for a sequence of
blockwise independent random variables obtained by Moricz [12]. In view of
Remark 3.1 (iii), we formulate Corollaries 4.1-4.5 assuming that p > 1.
COROLLARY 4.1. Let {It;,, n ~ 1} be a sequence of mean 0 random elements
in a real separable Rademacher
type p (1 < p ~ 2) Banach space and let
{bn, n ~ 1} be a nondecreasing sequence of positive constants such that (3.1)
holds. If {It;,, n ~ 1} is blockwise independent with respect to the blocks
{[2k-1, 2k), k ~ 1} (or, more generally, with respect to the blocks {[,Bk' ,Bk+ 1),
216
A. Rosalsky
k ~ 1}, where 13k=
L qk-l J
and Le Van Thanh
for all large k. and q > 1) and
co
(4.1)
if
E IIViIIP
i~l~<oo,
then (3.3) holds.
Proof. Recalling (2.2) and (2.3), we infer that the assumption (4.1) ensures
that (3.2) holds. The conclusion follows directly from Theorem 3.1. l1li
COROLLARY 4.2. Let {~, n ~ 1} be a sequence of mean 0 random elements
in a real separable Rademacher
type p (1 < p ~ 2) Banach space and let
{bn, n ~ 1} be a nondecreasing sequence of positive constants such that (3.1)
holds. If {~, n ~ 1} is blockwise independent with respect to the blocks
{[13k, 13k+ 1), k ~ 1}, where 13k= L 2k/(Logk)'"
J for all large k and a > 0, and if
~ EIIViIIP(L
(4.2)
i~2
~
L
og
')IX(p-l)
og z
<
00,
then (3.3) holds.
Proof. Recalling (2.4), we see that the assumption (4.2) ensures that (3.2)
holds. The conclusion follows directly from Theorem 3.1. l1li
COROLLARY 4.3. Let {~, n ~ 1} be a sequence of mean 0 random elements
in a real separable Rademacher
type p (1 < p ~ 2) Banach space and let
{bn, n ~ 1} be a nondecreasing sequence of positive constants such that (3.1)
holds. If {~, n ~ 1} is blockwise independent with respect to the blocks
{[13k, 13k+ 1), k ~ 1}, where 13k= L 2k'"J for all large k: and 0 < a < 1, and if
IEll ;II
P
(4.3)
i=l
(Log
1
i)(1X-
-1)(p-1)
<
00,
bi
then (3.3) holds.
Proof. By (2.5) the assumption (4.3) ensures that (3.2) holds. The conclusion follows directly from Theorem 3.1. l1li
4.4. Let {~, n ~ 1} be a sequence of mean 0 random elements
in a real separable Rademacher type p (1 < p ~ 2) Banach space and suppose
that {~, n ~ 1} is blockwise
independent
with respect to the blocks
{l Lk J, L(k+ 1) J), k ~ 1}, where al > 1. Let a2 ~ ail. If
COROLLARY
IX1
1X
1
IE ".;"
P
(4.4)
i= 1
where q
= (p (al a2 -1) + l)/al'
<
00,
t
then
1· I;=ln
Hfl
n~oo
Vi
IX
2
=
0 a.s.
Strong
Proof.
law of large numbers
in Banach spaces
217
Recalling (2.6), we obtain
P
~ E II ViII ( (i))P-1 ~ C ~ E IIVillP i(p-1)/cq
L.
'a2P
tp
"'"
L.
'a2P
i=1 1
i=1 1
=
C
IE ".;IIP <
and the conclusion
Remark
that
!X2
:::;
q
00
(by (4.4))
1
i= 1
follows from Theorem
3.1.
III
4.1. Apropos of the constant q in Corollary 4.4, it may be noted
<
!X2
p.
Counterparts to Corollaries 4.1-4.4 can easily be given when {~, n ;:::1} is
blockwise p-orthogonal with respect to a sequence of blocks. We content ourselves with only presenting the counterpart to Corollary 4.1. Corollary 4.5 is an
extension of the classical SLLN resulting from the celebrated RademacherMensov fundamental convergence theorem for sums of orthogonal random
variables (see, e.g., Revesz [16], pp. 86-87).
COROLLARY 4.5. Let {~, n ~ 1} be a sequence of mean 0 random elements
in a real separable Rademacher
type p (1 < p :::;2) Banach space and let
{bn, n ~ 1} be a nondecreasing sequence of positive constants such that (3.1)
holds. If {~, n ~ 1}is
blockwise p-orthoqonal
with respect to the blocks
k
1
k
{[2 - , 2 ), k ~ 1} (or, more generally, with respect to the blocks {[13k, 13k + 1),
k ~ 1}, where 13k = L q'c-1 J for all large k and q > 1) and if
E IIViIIP
i~1 bf(Logl)
00
(4.5)
.P
<
00,
then (3.10) holds.
Proof. Recalling (2.2) and (2.3), we infer that the assumption (4.5) ensures
that (3.9) holds. The conclusion follows directly from Theorem 3.3. III
We close by presenting four examples. The first example illustrates Theorem 3.1 and Corollary 4.1.
EXAMPLE 4.1. Let {~, n ~ 1} be a sequence of independent mean 0 random elements in a real separable Rademacher type p (1 :::;p :::;2) Banach space
and suppose that
I
i=1
00
(4.6)
EllmW <
'ap
1
00
2m:::; n < 2m+1,
{~, n ~ 1} is blockwise independent
Let
5 -
~
=
PAMS 27.2
~-2m+1'
for some
!X
> O.
m ~ O. Then E~ = 0, n > 1, and
with respect to the blocks {Ctc-1, 2k),
218
A. Rosalsky
and Le Van Thanh
k ~ 1}. Now, using (4.6), we obtain
~ E IIViIIP = ~ 2m~
1..J '(J.p
1..J
1..J
i=l
1
m=O i=2m
1
11U'i-2 +lIIP
E
m
'(J.p
1
3.1 (or by Corollary 4.1),
Thus, by (2.2) and Theorem
,,~
l7;
· 1..Jl=l(J. l
1Hfl
n->
=
n
00
Finally, we note that, by Theorem
,,~
a.s.
1 of Adler et al. [2J, we also have
JIJf;
1..Jl=n(J.
1
l
li
n~
°
=
°
a.s.
The second example shows that Theorem 3.1 can fail if the series in (3.2)
diverges. More specifically, Example 4.2 shows that we cannot replace (3.2) by
the weaker condition EII~IIP(cp(n)t-l/b~
= 0(1).
°
EXAMPLE4.2. Let {~, n ~ 1} be a sequence of independent mean
random elements in a real separable Rademacher type p (1 ::::;;
p ::::;;2) Banach space
where
P{II~II
1
= 2n-1}
= 1+Logn
'
Let lI;, = ~-2m+l'
2m ::::;;n < 2m+\ m
n ~ 1, and {~, n ~ 1} is blockwise
{[2k-1,
2k), k ~ 1}. Now V2m+l-1 =
a sequence of independent random
P{~
1
1 +Logn'
= O} = 1
n ~ 1.
~ 0, and let bn = n, n ~ 1. Then Ell;, = 0,
independent with respect to the blocks
W2m, m ~ 0, and so {V2m+l-1,
m ~ o} is
elements with
P {V2m+l-1
=
o}
m
= --1'
m+
m~O.
Then, recalling (2.2), we obtain
~ EIIViIIP(
.1..J
l=l
b-P
l
(.))P-l
cp
1
~
s-:
~ EIIV2m+1-11IP
1..J (2m + 1-l)P
m=O
00
(2m + 1_ l)P
= m~o(m+1)(2m+l-1)p
00
=
m~om+1
1
= 00,
219
Strong law of large numbers in Banach spaces
and so (3.2) fails. Moreover,
by the Borel-Cantelli
since
lemma we have
Thus,
1 -1'
- Imsup
2m+1-111 ~ u
IIVm+l
II~II
1 -...;;::
Imsup
2
m-+co
____
l'
~ nn sup
11-+
co
-
n
n-+co
III;= ~II+ uim
1
n
III;~:1~II a.s.
sup
n-
n-+co
implying
III;=b ~II_- l'im sup III;=
· sup
1im
1
n-+co
1 ~"
n
n-r
n
co
-:/ -1 a.s.
2
'-
Thus (3.3) fails. We also note that for all large n, writing 2m
again recalling (2.2), we get
E
II~IIP(
bP
n
((J
~
n < 2m+1 and
= E 11~-2m+ l11P
())P-l
n
w
(2(n-2m+ 1)-1t
nP (1 + Log(n-2m+ 1))
+
(2(2m+1-1-2m
1)-1t
~--'------------'--mp
m 1
m
2 (1 Log(2 + -1-2
1))
+
~ --
2P
m+1
+
as n
-+ 0
-+ CIJ.
The third example shows that the Rademacher type p hypothesis in Theorem 3.1 cannot in general be dispensed with. Example 4.3 concerns the
real separable Banach space 11 of absolutely summable real sequences
v = {Vj,j ~ 1} with norm Ilvll= I:1IvJ The element of 11having 1 in its nth
position and 0 elsewhere will bel denoted by v(n), n ~ 1.
EXAMPLE4.3. Consider the real separable Banach space 11. It is well
known (see, e.g., Adler et al. [1]) that 11 is not of Rademacher type p for any
1 < p ~ 2. Define a sequence of random elements {~, n ~ 1} in 11 by requiring
the {~, n ~ 1} to be independent with
p {~
=
v(n)}
=
p {~
= -
v(Il)}
= 1,
n ~ 1.
220
A. Rosalsky
and Le Van Thanh
Let ~ = ~ _2m + 1, 2m ~ n < 2m + 1 , m ~ 0. Let 1 < p ~ 2 and let b; = nIX,n ~ 1,
where
< a ~ 1. Then E~ = 0, n ~ 1, and {~, n ~ 1} is blockwise independent with respect to the blocks {[2k-1,
2k), k ~ 1}. Now, by (2.2),
«:
IEII;IIP
(cp(oy-1 = I ':P <
b
i=1
00
i=11
i
since ap > 1, and so (3.2) holds. Next, for MEN,
+1- 1
+1- 1
+1- 1
Ii=1 Vi=I m=O i=2I Vi=I m=O i=2I
2M
M
2m
M
2m
m
M
2M
I I
=
m
2m
Wii =
m=O i=1
2M
M
I
I
I
Wii =
i=1 m=iLogil
1) Wii,
(M -iLogil+
i=1
and so
III~~1+1-1
Viii III~~1(M -iLogil+
2M
=
Consequently,
1) Wii'l
(2M+1_1)IX
b 2M+1-1
Ii=1 (M -iLogil+
1).... 2M
~ 2M+1
(2M+1_1)IX
1
="2
a.s.
(3.3) fails.
The final example, which is a modification of an example of M6ricz [12J,
shows apropos of Corollary 4.1 that under its hypotheses the series
1 Vi/bi
can diverge a.s. Consequently, the conclusion of Corollary 4.1 (or Theorem 3.1)
cannot in general be reached through the well-known Kronecker lemma
approach for proving SLLNs.
I;=
4.4. Let the underlying Banach space be the real line and let
p = 2. Let {Xn, n ~ 1} be a sequence of independent mean
random variables
such that P {X 1 =I- O} = 1 and
1 EX~ < 00. Define for n ~ 1
°
EXAMPLE
I:=
v: _
n -
{nXt!(l
nXn,
+Logn),
LognEN,
Logn¢N,
and let b; = n, n ~ 1. Then E~ = 0, n ~ 1, and {~, n ~ 1} is blockwise independent with respect to the blocks {[2k-1,
2k), k ~ 1}. Now
Ev:2
1
I -+ = EXt m=o(1+m)
I
n=1 bn
C1)
C1)
C1)
2
+
I
n=3
EX~ <
00,
Logn¢N
and so, by Corollary
4.1, (3.3) holds. However,
n
X1
i=1
1 +LOgl
I
LogieN
[Logn ]
1
.= X1 I -m=O
1 +m
diverges a.s. as n ~
00,
221
Strong law of large numbers in Banach spaces
and by the Khintchine-Kolmogorov
Teicher [6], p. 113)
convergence theorem (see, e.g., Chow and
n
I
Xi converges
a.s. as n ~
CIJ.
i=3
Logi¢N
for n ~ 3
Consequently,
±
Vi =
i = 1 hi
±
i= 1
LogieN
Xl
1 + Log
.+
1
±
Xi diverges a.s. as n ~
CIJ.
i=3
Logi¢N
Acknowledgements. The authors are grateful to the referee for carefully
reading the manuscript and for offering some very perceptive comments which
helped them improve the paper. The authors also thank Professors Nguyen
Duy Tien (Viet Nam National University, Ha Noi) and Nguyen Van Quang
(Vinh University, Nghe An Province, Vietnam) for their interest in our work
and for some helpful and important remarks. The research of Le Van Thanh
was supported by the National Science Council of Vietnam.
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Department
of Mathematics
Vinh University
Vinh City, Nghe An Province
Vietnam
E-mail: [email protected]
Department of Statistics
University of Florida
Gainesville, Florida 32611, USA
E-mail: [email protected]
Received on 7.10.2005;
revised version on 31.5.2006