805
Internat. J. Math. & Math. Sci.
Vol. i0 No. 4 (1987) 805-814
STRONG LAWS OF LARGE NUMBERS FOR ARRAYS
OF ROWWISE INDEPENDENT RANDOM ELEMENTS
ROBERT LEE TAYLOR
Department of Statistics
University of Georgia
Athens, GA 30602 U.S.A.
TIEN.CHUNG HU
Department of Mathematics
National Tsing-Hua University
Hsin-chu, Taiwan R.O.C.
(Received November 18, 1986)
ABSTRACT.
Let
{Xnk
be an array of rowwise independent random elements in a separable
Banach space of type p
+
6
with
0 for all k, n. The complete convergence
n
to 0,
! P < 2, is obtained
EXnk
(and hence almost sure convergence) of n -I/p
{Xnk
array {Xnk
when
k= Xnk
consists of i.i.d,
converges completely to 0 if and only if
KEY WORDS AND PHRASES.
2p...
EIIXIIII
<
k=l Xnk
.
Random elements, Strong laws of large numbers, Complete Con-
vergence, Rademacher type p
+
6 spaces.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODE.
1.
EIXI 2p
=. When the
-i/p n
random elements, then it is shown that n
are uniformly bounded by a random variable X with
60B12
INTRODUCTION AND PRELIMINARIES.
Let (E,
space.
II
II)
Let (,A
be a real separable Banach space.
A random element
X in E is a function from
into
E
p) denote a probability
which is A-measurable
pth
with respect to the Borel subsets
8(E). The
absolute moment of a random element
p
The expected
X is
where E is the expected value of the random variable
IIXII p.
EIIXII
value of X is defined to be the Bochner integral (when
EIIXII
=) and is denoted by EX.
E.
The concepts of independence and identical distributions have direct extensions to
A separable Banach space is said to be of
(Rademacher.)
type p,
<_ p <_ 2,
if there
exists a constant C such that
n
E
--k:l
n
p < C
Xk
k:l
E[[Xk[[
p
th
moments.
for all independent random elements X I,
with zero means and finite p
2. Every
of
type
is
Every separable Hilbert space and finite-dimensional Banach space
p
and L spaces are of type
while the
separable Banach space is at least type
Xn
EP
min{2,p} for p >_ I.
Throughout this paper
{Xnk:
<_
k < n, n
>_ i}
will denote rowwise independent
random elements in E such that
EXnk
and such that
{Xnk}
0.
for all n and k
(1.1)
are uniformly bounded by a random variable X with
EIXI 2p
for some l<_p <2.
(1.2)
R.L. TAYLOR and T.C. HU
806
array{%nk}
Recall that an
of random elements is said to be uniformly bounded by a
random variable X if for all n and k and for every real number t > 0
P[IIXnkll
Note that
i.i.d, random elements are
IXl
>t] ! P[
t]
>
(1.3)
uniformly bounded by
llXllll.
The major results
of this paper show that
1
n
0
nl/----p k=l Xnk
(1.4)
completely
where complete convergence is defined (as in Hsu and Robbins [i]) by
n=l
xnk=l Xnkll
P[
n
> e
]
(1.5)
for each e > 0.
Erds [2] showed that for an array of i.i.d, random variables {Xnk }, (1.4) holds
<. Jain [3] obtained a uniform strong law of large numbers
for sequences of i.i.d, random elements in separable Banach spaces of type 2 which
would yield (1.4) with p
for an array of i.i.d, random elements {Xnk in a type 2
space. Woyczynski [4] showed that
if and only if
EIXI112p
1
n
0
nl/--- k=l Xk
for any sequence
with EX
n
EIXI p
.
{Xn}
(1.6)
completely
of independent random elements in a type p
+ 6,
1
!P
< 2 and 6 >
0,
0 for all n which is uniformly bounded by a random variable X satisfying
<
Mricz, Hu and Taylor [5] showed that Erds’ result could be obtained by
replacing the i.i.d, condition by the uniformly bounded condition (1.3). In addition,
they showed that Jain’s result for i.i.d, random elements with p 1 did not require
the space to be type 2 but held in all separable Banach spaces.
is established in type p
+ 6 spaces, 1
independent random elements.
For
!p < 2 and 6 > 0, for uniformly bounded row-wise
i.i.d, random elements in
shown that (1.4) holds if and only if
In this paper, (1.4)
EIIXIIII
2p <
.
type p
+
6 spaces, it is
Thus, no sharper moment conditions
are possible.
2.
MAJOR RESULTS.
Many authors (starting with Beck [6] have related the strong law of large numbers
for non-identically distributed, independent random lements in separable Banach spaces
to the necessity of the space being of type p+6 for l<p < 2 and some 6
0. Consequently, attention is restricted to type p + 6 spaces in this paper. Three lemmas
will be used in obtaining the major results.
Lemma
They are stated here without proof.
1 is in most textbooks while Lemma 2 is accomplished using integration
Lemma 3 is in Woyczynski [4].
with r
by parts.
LEMMA I.
For any r > I,
EIXI r
n=l
nr-ip[
More precisely, r2 -r
n=l
<
IXI
ElXlr <
<
if and only if
nr-Ie[ IXI
>
>n]
<
.
n]
+ r2 r
n
n=l
r-i
e[
IX
>
n].
STRONG LAWS OF NUMBERS FOR INDEPENDENT RANDOM ELEMENTS
LEMMA 2.
>_ i, then for any p
If r
IXII [
E
0
i/p
tr-i P[ IXl
< r
IXl<n 1/p ]
807
>
t]dt
0
and
IX
E
LEMMA 3.
(i) E
I
[
Let
<_ p <_ 2
is of
nlIpp [ Ix
IXl>n llp],
and q
>_
] +
>n
IXl
P[
llp
n
t]dt.
The following properties are equivalent:
i.
type p.
(ii) There exists a C such that for all independent random elements X
in
E
%
with
0, k
p
+
6 space,
<_ p
If
<
{Xnk
Xn
n,
llk=Ix Xkll q_<
E
THEOREM 4.
i,
fIXkll p
C E
k
q/p
is an array of rowwise independent random elements in a type
2 and 6
0, which are uniformly bounded by a random variable X such
that (I.i) and (1.2) holds, then
n
X
nl/P k=l nk
PROOF.
0 completely.
Define
Ynk Xnkl
Then, by Lemma i (with r
[l[Xnkll
< n
n, n > i.
< k
I/p ]
(2.1)
2),
n
n=l k=l
X nP[
n=l
IX[
X nP [
n=l
Ixl p
> n
n
:
Xnk Ynk
P[
n=l k=l
> n I/p
..llXnk
]
l/p]
> n
] !
2EIXI 2p
<
.
Next, for any e > O,
n
n
p[[[
k=l
n=l
<_ X P[ U
k=l
n=l
<_
.
n
7.
n=l k=l
n-7 kl Ynkll
Xnk
[Xnk + Ynk
P[Xnk + Ynk ]
e
]
]
"
Therefore,
I
n
n
I xnk
n
I/p
kl Ynk 11
0
completely,
and it sufficies to prove that
n
Ynk
0 completely.
(2.2)
R.L. TAYLOR and T.C. HU
808
To this end, let
(k=l,2
Znk Ynk EYnk
Then for
<_ q <_ 2p
).
n=1,2
n;
Hider’ s inequality that
I/q
I/q
it follows by
(gllZnkllq)
2(EllYnkllq)
2(gllYnkll2P) I/(2p) <_ 2(EIxI2P) I/(2p)
<_
<_
so that
E[[Znk[[q _< 2q(E[xI2P) q/(2p)
<_ 22P(I + EIX [2p) CI say.
(2.3)
Furthermore,
+
Ynkll
<_
Znk
<_
EYnk
2nl / p.
(2.4)
Following the techniques of Taylor [7] in expanding a high power of a sum, let
r
p+6 and
be chosen so that
s
r
It is readily seen that E
llZnkll
<
,
Znkl[j
< C
E
n
E
(-- 1)-i
P r
is an integer and v >
(IIk=l
so that, by Lemma 3,
I[Znkl[r
(kl)s
x
c
j.
][Z n
E
J
ks
kl
(k I,
where the sum is extened for all s-tuples
(2.5)
ks
I, 2,
k.3
with
The general term to be considered then will have
ql of the k’s
$I’
qm
of the k’s
D I,
r of the k’s ;
r
where
r
<_ rqi
<
2p,
rrj
of the k’s
(2.6)
Sm;
2p, and
>
n for each
(2.7)
m
l.
i=l
Clearly, qi
I.
qi +
r.
52.
(2.8)
s.
3
j=l
Then, using (2.3) and (2.4), we can conclude that
m
E
i=lll Zni
rqi
I
j--i
II
ZnDJ
rr
j
J
ilrrj-2p
<
C
(2nl/p)rrj -2p
j=l
_m+
2
1
G
jl (rrj
2p)
. (rrj/p)-2
nJ=l
809
STRONG LAWS OF NUMBERS FOR INDEPENDENT RANDOM ELEMENTS
X (rr. Ip)-2
C
2
j=l
n
X (rr Ip)-2
=C
2
n
j=l
say.
Combining all possible terms of form (2.9), we can write
n
! c3
ql
C
r
qm;rl
say,
s.
We distinKuish two cases according to t
Case t
r)
and -tuples (r
0 or
N)
’m; HI
2
such that
0 may also occur),
and n and C is a constant independent of n.
integers between
it !
qm
(ql
(2.7) and (2.8) are satisfied (the cases m
tuples (6
is extended over all (m + )
Conditions
while
j=l
i
r,
qm;rl
ql
is extended over all m-tuples
where
q
S
3
ql
rrj)
m
x
m;ql
r 61
qm;rl
(2.o)
Let m +
of different
Obviously,
i.
or t
2. By (2.9)
S
ql
r
qm;rl
<_ C 2
! C2
n
n
n
m;rl
61
X
j=l
(rrj/p)
2
+
-
j=l
(rrj/p)
2
t
(2.t)
Now, the power to which n is raised here can be estimated by means of (2.8) and qi
as follows
P j=l
I
i
P
rr.
2
+
rqi
)
o
rs
52
i=l
rm
P
P
3
Z
t
-p
2(t
t
2(t
m) + t
m) + t
m( 2).
(2.12)
We distinguish two further subcases according to m
Subcase m
Thus, m
t.
By assumption
s and, by
(2.5).
! p
2.
Also, qi
t or m
!
t
i.
for each i.
R.L. TAYLOR and T.C. HU
810
(2.13)
I.
t
Subcase m
m
0.
Then t
i.
2 in the particular case where
m
and even t
m
Thus, again
t+m
Now
we turn to
Case t
In this case necessarily m
i.
0 and
n
S
ql’
S
,r
’qm;rl
7.
k=l
O ;s
i, consequently r
s and
EllZnkll rs.
Using Lemma 2, we obtain that
n
7.1
<2
v
n=l n
=I EllYnk II
7.
n=l
<_2
2
Letting t
n
1/p
s
n
7.
7.
n=l
0
k=l
.
v
n
n
t/p
-
P[llXnkll
t
1/p
v n
t
e[
IXl
1/v and
applying Lemma I (with r
v
<_ 2
7.1
2
i
k
v
.i
2v+l
2
7.
n
n=l
7. nP[
n=l
i
P[
IX[
Is- i/u X[ p
fl s- 2p/ EIX
>
2),
>
t]dt
t]at.
it follows that
>
nl/P sl/ ]ds
>
n]ds
(2.15)
2p ds
2p
v+l
Using Markov’s inequality, (2.7) and (2.10)
z PElf
7.2(e)
n=l
<
Znkll
n
z
n=l (n / P)
(2.15) we have, for any e
>
e]
Znkll
E
n
Z
n=l
1/p
:n
kl Ilznk
)
> 0,
-
811
STRONG LAWS OF NUMBERS FOR INDEPENDENT RANDOM ELEMENTS
<_
CC3
E
+
n=l
n
[nl
c2
n v/P
CC 3
(r
(t)
r)
v
(t)
t=2
n
r
qm;rl
ql
s
7.
7.
t=2
(t)
r
qm;rl
ql
p
-t-m( [
7.n
n=l
7.2(e)
<
(t)
.
]
-t-m( r
P
qm
(ql
m+
with Conditions (2.7) and (2.8) such that
2)
P
means that the sum is extended over all m-tuples
terms in each of
we have
s
+ C2
[7.1
e
where
k=l EllZnkll
v/p
n
2)
]
and -tuples
Since the number of
t.
-I, for every
is finite and the exponent of n is less than
e > 0,
Thus, we have proved that
n
lln- k=l Znkll
0
completely (n
).
In order to prove (2.2), we need to establish
n
3
nl n- k--i llEYnkll
<
By (2.1),
To achieve this goal, we will proceed as follows.
I
Ynk Xnk
[
Xnk <_. n
Xnk- Xnk I [
p
]
llXnkll
Since
EXnk
(2.16)
"
n
i/p
0, hence
lIE Ynkll
<_
E(llXnkll
i/p])"
I
[
llXnkll
> n
Thus, using Lemma 2,
Letting t
n
k E ([[ Xnk
I
i/p
n
.=
7.n (nl/PP[ llXnkll
n=l
<
n
X:3 <- n=l
I
llXnk[l
[
n
I/p]
+
k=l
X (n P[
n=l
s and applying
3
7. n
<- n=l
P[
<_ 2
EIXI 2p
+
IX[
n
I/p]
+
>
nl/p] j
[,
P[
nllP
n
IX
P[
n
/p
>
llXnkll >t ]dt)
t]dt ).
Lerra 1, we can conclude that
[Xl p
>
n] + 7. n
P[
n=l
X P[
n=l
l,s -IX[ p
>
n]ds
IX[
> n i/p
s]ds
R.L. TAYLOR and T.C. HU
812
s- 2p EIX
4PCI
2p-i
2p ds
EIxI2P’
proving (2.2) through (2.16), and thereby completing the proof of Theorem 4.
Note that if sup
nk
%ukP
EIXI 2p
Therefore, Corollary 5 follows.
< =.
Let E be a type p+ 6 separable Banach space for
COROLLARY 5.
If
0, then there exists a r.v. X such
>
for some
are uniformly bounded by X and
{Xnk
that
EllXnkll2P+=<"" ,
E[[Xnkll2P+
=
for some
///
< p
2 and 6
0.
0, then
n
[[n- k=l Xnk[[
0 completely.
p
+
For type
6 spaces, Taylor [7] obtained
0 completely
k=l ank Xnk
where
{Xnk
with max
r
I
is uniformly bounded by X with
0(n-r)
ank
nd Theorem 4
E[XI l+I/r
<
and
{ank
In the special case of uniform weights
are Toeplitz weights
ank
can be thought of as an extension of this result.
Theorem 4 to infinite arrays and
Keneral
weights
,
(2.17)
i < k < n, then
Extension of
are possible but the detailed
{ank
However, it will be shown next
cannot be reduced in Theorem 4. In particular,
0, it will
for an array {Xnk} of i.i.d, random elements in a type p+ 6 space with
2p <
if
holds
and
only
if
be shown that the SEEN
verification of their proofs are not included here.
2p <
that the moment condition EIX[
Let
THEOREM 6.
lip
{Xnk}
< 2 and 6 > 0, with
be an array of i.i.d, random elements in a type p
EXII
i
nl/P
{Xnk}
0.
Then
k=In Xnk
EIIXIIII
0
From Theorem 4, we know that E
PROOF:
is uniformly bounded by
Now, assume that (2.18)
[[Xll[[.
holds.
Since
2p <
IIXIIII 2p
implies (2.18) since the array
are i.i.d., for every n and > 0
n
O,
xkll
which says
1
n
kiXkk
As a consequence,
6 space,
(2.18)
completely.
{Xnk}
+
if and only if
n
By (2.18), for every e
EXII
.
E[[Xll[[
0 a.s..
>
]
<
-,
(2..)
813
STRONG LAWS OF NUMBERS FOR INDEPENDENT RANDOM ELEMENTS
nl/P
Let e
I.
In_l n-l)i/P k=l xkk
n
n-i
-n i
xnn
Xkk
EIIXIIII p
+ 2
<
P [
n;1
+ 2
p
fIX
n]
II Xnnll
P[
n=l
a.s..
I) and the Borel-Cantelli lemma that
(with r
It follows from Lemma
0
n
]
.
<
n
Hence
llXllll
nP[
p
n]
O.
(2.20)
1/p ]
(2.21)
By (2.18),
. Xnk
n
P[
< n
k=l
Therefore, from (2.20) and (2.21) there exists N such that if n > N then
n
llx
nP[
P[
and
p <
n]
z
llk=l xnk
"
Next, define the events
Ank [l<i<kmax llXnil[ < 2nl p, llXnkll 2nl p,
(2.22)
n
and
lli=im Xnill
< n
1/p ]
ik
(k
i, 2
).
i, 2
n: n
n
Clearly,
{Ank:
for each n
k
n} are disjoint subsets of the event [
1, 2,’
[[kl xnk[[
n
e[
k=z llXnk
Z P[
k--I
>
-
l/p]
n
n
n
> n
A familiar reasoning yields that
i, 2
e[
EX
k=l nk
>
] >_
2nllp ]
>
XI I
’f[k--1
p
li=l
[
2nl/ P ]
llxni
n[
L
llXnkll > 2nllp ]
" P[
k=l
P(Ank)
k=l
>
2nl / p ]
p[
i
P[
i
n
i=z x
< n
i+k
Xnill
xI
< n
]
P[ U
i=l
<
(n-
I/p ] -nP [
[llXnill
Ix
> 2n
2
l/p] ]
)
n Ip ] 9"
Hence, by (2.22), for n >_ N,
n
p[
Therefore,
[[--p
k Xnk[[
n
Z nP[
n=l
IIXIIII p
>
>
nP[
] >_
2Pn ]
<
.
llXll[l p
>
2pn ]"
Thus, Lemma I yields
EIIX IIIIzp
<
.
III
814
R.L. TAYLOR and T.C. HU
CONCLUDING REMARKS.
i.
It should be noted that the case p
space (cf:
in Theorem 6 is obtainable in a type
Theorem 4 of Hu, Moricz and Taylor [5]).
In which case type
+
6 is not
needed.
For sequences of independent random elements which are uniformly bounded by a
p <
(1.6) holding necessitates the space being of type
random variable X with
p + 6 (cf: Woycyznski [4] and Maurey and Pisier [8]). Thus, the necessity of type
2.
EIXI
p
+
,
6 follows for Theorem 4.
3.
Theorem 6 shows that Theorem 4 is the best possible moment condition when no
conditions on possible relations between the rows of the array are assumed.
-i/p n X
4. In [4] it is mentioned that n
0 a.s for i i d random elements
k
p
<
0 and
{Xn with EX
apparently is equivalent to the space being of type
p. Thus, it is interesting to conjecture whether Theorem 6 remains valid for only
k=l
EIIXIII
type p spaces
!p
< 2.
Certainly, the "if part" is true for type p spaces, and Remark
indicates that it is true p
ACKNOWLEDGEMENTS.
1.
This research was supported in part by the Air Force Office of
Scientific Research under Contract No. F49620 85 C 0144 while the first named author
was at the Center for Stochastic Processes, University of North Carolina, Chapel Hill,
N.C. The research for the second named author was mainly completed while at the
Department of Statistics, University of Georgia, Athens, GA.
REFERENCES
i.
2.
3.
4.
5.
6.
7.
8.
HSU, P.L., and ROBBINS, H. (1947). Complete Convergence and the Law
of Large
Numbers. Proc. Nat. Acad. Sci. U.S.A., 33, 25-31.
ERDOS, P. (1949). On a Theorem of Hsu and Robbins. Ann. Math. Statistics 20
--’
286-291.
JAIN, N.C. (1975). Tail Probabilities for Sums of Independent Banach
Space Random
Variables. Z. Wahr. v. Geb., 33, 155-166.
WOYCZYNSKI, W.A. (1980). On Marcinkiewicz-Zygmund Laws of Large Numbers in
Banach
Spaces and Related Rates of Convergence. Prob. and Math.
Statist., l, 117-131.
HU, T.C, MORICZ, F.and TAYLOR, R.L. (1986). Strong Laws
of Large Numbers for Arrays
of Rowwise Independent Random Variables. Statistics
Technical Report 27,
University of Georgia, March, 1986.
BECK, Ao (1963). On the Strong Law of Large Numbers.
Erodic Theory. Academic
Press, New York, 21-53.
TAYLOR, R.L. (1982). Convergence of Weighted Sums of Arrays
of Random Elements in
Type p Spaces with Application to Density Estimation. Sankhy
a, 44, 341-351.
MAUREY, B. and PISIER, G. (1976). Series des Variables Alatoires
Independantes et Propriets Geom4triques des Espaces de BanachVectorielles
Studia Math
5_8, 45-90.