69
COMPLETE CONVERGENCE FOR WEIGHTED SUMS
OF ARRAYS OF RANDOM ELEMENTS
ROBERT LEE TAYLOR
Department of Mathematics and Statistics
University of South Carolina
Columbia, S. C. 29208
(Received July 22, 1982)
ABSTRACT.
{Xnk:
Let
k,n
1,2
be an array of row-wise independent random
elements in a separable Banach space.
Voo
real numbers such that
An
V
kk=l
2
/-k=l
lank
{ank:
Let
be an array of
+ where
voo
Ln=l exp(-a/An <
-< 1 and
Voo
The complete convergence of
ank.
1,2
k,n
l’k=l ank
Xnk
for each c e R
is obtained under varying
moment and distribution conditions on the random elements.
In particular, laws of
large numbers follow for triangular arrays of random elements, and consistency of the
kernel density estimates is obtained from these results.
KEY WORDS AND PHRASES.
Weighted Sums, Random Elements, Laws of Large fTumbers, subGaussian, Complete Convergence, and Kernel Density Estimates.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
I.
Primary 60B12, Secondary 60B11.
INTRODUCTION AND PRELIMINARIES.
Wei and Taylor [i] obtained the
convergence of
[k=l
ank
Xk
in probability and
with probability one by assuming tightness (given e > 0 there exists a compact set K
such that
sp
P[X
n
K
e and uniformly bounded
E
the sequence of random elements
{X
n
pth
(p > i) moment conditions on
in separable Banach spaces.
Howell and Taylor
[2] proved Marcinkiewicz-Zygmund type weak laws of large numbers for the weighted sum
Vn
kk=l ank
Xnk
of arrays of random elements
tain geometric conditions.
convergence, is
obtained
for
{
Xnk
in Banach Spaces satisfying cer-
In this paper a stronger mode of convergence, complete
Lk=l ank
Xnk
in separable Banach spaces with less re-
strictive conditions than the results of Wei and Taylor and without assuming geometric
R.L. TAYLOR
70
The format for these results was motivated in part by the
conditions on the spaces.
form of the kernel density estimates (which are weighted sums of arrays of random
elements) and by the relative absence of laws of large numbers for arrays of random
elements.
Let E denote a real separable Banach space with norm
If
Let (,
A random element X in E is a function from
denote a probability space.
A, P)
into E
The expected value of
which is A-measurable with respect to the Borel subsets of E.
X is defined to be the Pettis integral (when it exists, see pp. 38-41 of [3]) and is
denoted by EX.
The p
th
moment of a random element X is
pected value of the (real-valued) random variable
E(
lXl p.
IxI
p)
where E is the ex-
The concepts of independ-
ence and identical distributions (i.i.d.) have direct extensions to E.
random elements {X
converge completely to the random element X if
is said to
n
n=iP[IIXn-XII > ] <
n=iP[l]Xn Ell > ]
> 0.
for each
<
,
A sequence of
Since
P[n$1[IlXn XII
> eli
complete convergence implies convergence with probability
one, but the reverse implication need not hold.
Because of the methods of proof,
complete convergence will be obtained for the results of this paper rather than (the
standard) convergence with probability one.
2.
The
Complete Convergence of Weighted Sums.
In this section complete convergence for weighted sums of arrays of random
Since no geometric conditions are assumed on the space, it is
elements is obtained.
easy to show that moments conditions alone will not suffice.
Thus, sigma compact
support or uniform compact integral conditions will be assumed on the distributions
of the random elements.
Throughout this section
elements in a separable Banach space E such that
each n.
Moreover,
{auk}
k=l lank
for each
> 0 where A
n
{Xnk} will denote an array of random
{Xnk: k e I} are independent for
will always denote an array of real numbers such that
I and
n=l
exp[-/An
<
(2.I)
2
k=l
auk"
First, the complete convergence of the weighted sums will be obtained when the
random elements are restricted to a compact subset with probability one.
This result
will allow consideration of the random elements which are truncated to compact subsets
COMPLETE CONVERGENCE FOR WEIGHTED SUMS
71
and the corresponding parts off the compact subsets.
THOREM I.
Let K be a compact subset of a separable Banach space E.
[Xnk:
is an array of random elements in E such that
e K]
P[Xnk
n and such that
i and
I.k=l ank Xnkll
PROOF.
points of K.
0 for all n and k, then,
EXnk
0 completely.
{f
there is a countable set S
Also, for each e > 0 there exists
{x
II
e K:
x
}’k ankl
Nxt, for h
K (wlog).
u
i=l
’P
{x
e K:
If i I[
with
’fm S such
[fi(x)I > }.
by on,,xty that
1
In
which separate
that
(2.2)
)k-- ank Xk
K wt
Hence, for each e > 0 and each n
probability one.
P[-i
-
> E}
i
{fl
m
II
{Xnk}
k -> i} are independent for each
It can be assumed that K is convex and symmetric and 0
the dual space E
If
l.k= 1 ank Xnkll
> g]
(2.3)
For each
fi, E[fi(Xnk)
with probability one.
0 for all n and k, and
{fi(Xnk)/F}
<
If fi I(sup If
x
If)
xK
Hence,
E[exp(t
or
fi(Xnk)
fi(Xnk))]
< e
F2t 2
are sub-Gaussian with parameters
(2.4)
-< 2 112 for each
i.
By Theorem 2 of
Chow [4],
i
for each f..
[k=l
ank fi (Xnk)
-
0
<
=.
completely
(2.5)
Combining (2.3) and (2.5), it follows that for each
n=l
PC
llk=l ank Xnkll
> c]
///
Using truncation to a compact subset and Theorem i, a strong law of large
numbers will be proved for triangular arrays of random elements.
THEOREM 2.
space E satisfying
Let
{Xnk}
be an array of random elements in a separable Banach
72
R.L. TAYLOR
(i)
{Xnk
(ii)
EXnk
(iii)
(iv)
k > i} are i.i d
for each n,
0 for all n and k,
’n=l E(IIXnl II2q)/nq <
for some q
2
i, and
given e > 0 there exists a compact subset K
lim sup
n
Then,
1
E(IIXnlII I[ Xnl K])
n
I k=l Xnk If
PROOF.
0
> 0 choose K
Given
such that
< e.
completely
compact, convex, and symmetric with 0
K
such
that
limn
by (iv) and (i).
sup
i
ank
in
i[ Xnk
for k
l[XnkL
Kg]
e
i,..., n and
and
ank
Znk Xnk
(2.6)
{Ynk EYnk}
IF_Xnk
(2.7)
K ]"
0 for k > n, it follows that
’n-l- exp[-<I/An ’n-l- exp[-an] <
Since
< E
Ke]
Define
Ynk Xnk
Using
n
k=l E(I IXnk ii
An
I
and
oo.
take their values in 2K,
=i P[IIk=l ank(Ynk
EYnk) II
>
]<
22q
E
(2.8)
by Theorem i.
For each n and k
Eli
IZnkl]
E
]Znkll 12q
_<
IIXnkl 12q
(2.9)
> 0 and each n
Thus, for each
n
P[[-1 .k=l([
[Znk[[
E[ [Znk [[)[
-<
(en/4)-2q
<
(en/4) -2q C
>
g
-]
EF’ln-k=l(IIZnkll EIIZnkIl)I2q]
I
n
q
Eli IZnll[
by Marcinkiewicz-Zygmund’s inequality (rf:
EI IZnlll 12q]
Corollary 2.1, Woyczynski F,5]).
(2.10)
73
COMPLETE CONVERGENCE FOR WEIGHTED SUMS
From (2.9), (2.10) and (iii),
n
’n=l P[I- k=l(i lZnkll EllZnkl I)i
<
where C
2
C2
’--1 E(][Xnl li2q)/n
is a constant independent of n.
q
>
(2.11)
<
Finally,
n
> ]
I Xnkll
n=l P[II Ln=
n
n=l P[IIi Ik= l(Ynk mYnk) > ]
+ ln=l P[]n k=l Znk
nk
+ In=l P[2 1n nk=l El Znk II > ]
l-EZnkll
lEYnkll
from (2.6), (2.8), (2.11) and the fact that
-<
El [Znkll
for
///
all n and k.
The form of Theorem 2 is very appropriate for applicat+/-ons in kernel density
estimation where
XI, ...,Xn
is a random sample from a distribution with unknown
probability density function f.
fn (t)
.n
i
’i=l
n
The kernel estimate for f is given by
K(
t-X.i
(2.12)
n
where K is an arbitrary kernel function which is usually assumed to be bounded,
nonnegative and integrable to i and where hn is a sequence of positive numbers tending
-
to zero which adjusts the height and spread of the kernel function.
elements
{Xnk}
Xnk
K(-----)
n
n
satisfied.
The random
in an appropriate Banach subspace of function space can be defined as
n
E K(
)"
h
Conditions (i) and (ii) of Theorem 2 are trivially
n
For complete convergence in the
E(F_o
I--n
Ll-norm,
t-X
t-X
(K( h I
n
(iii) is easily satisfied since
E
I
K(-hn ))Idt)mq
R.L. TAYLOR
74
E(f_ K(s-XI/hn )ds + f
-<
E(I + i)
In the sup-norm, E
2q
2
(K)]2q
_<
t-X
i
h
EK
n
I
h
1
dt) 2q
n
.oo hn2 )-q
n=l
n
<
for some
0,
f (x)
is achieved with traditional techniques, and (complete
kernel density estimate in the
n
and thus
Finally, when hn
(-----)
n
t-Xl
2q
IIXnlIl2q [_2n bound
q > 1 is sufficient for (iii).
E K(
Ll-norm
convergence) consistency of the
or the sup-norm is obtained from Theorem 2.
The verification of (iv) depends on the particular Banach space
ular, (iv) is implied by tightness and uniformly bounded r
easy to see that tightness and uniformly bounded r
for Condition (iv) of Theorem 2 even when E
th
th
moments
In partic-
(r > i).
It is
moments (r > i) are not necessary
R.
The complete convergence for weighted sums of arrays of random elements will
Only the parts of the proof which differ from the proof of
be obtained in Theorem 3.
Theorem 2 will be presented.
First, let
X
denote the essential supremum of a non-
negative random variable X.
Let
THEOREM 3.
{Xnk}
be an array of random elements in a separable Banach
space E satisfying
(i)
[Xnk
k >- i} are independent for each n,
(ii)
EXnk
0 for all n and k,
(iii)
sup
n,k
(iv)
{ {Xnk{{
given
F < oo, and
> 0 there exists a compact set K
lim sup
"k=l
n
ankIE(I Xnkl II[Xnk
K
])
such that
<
g.
Then
Ilk= Iank Xnkll
OUTLINE OF PROOF.
0
completely.
Steps (2.6), (2.7), and (2.8) in the proof of Theorem 2
75
COMPLETE CONVERGENCE FOR WEIGHTED SUMS
remain the same.
bounded by
r
{l[Znkll El [Znk[I} have zero means and are
E
Hence, {IIznkll
IIZnkl I} are sub-Gaussian
The random elements
with probability one.
I/2
with parameters < r2
and
Ik= I ank(IlZnkll
El
IZnklI)]
(2.13)
0
Thus, the proof is completed by following the
completely by Theorem 2 of Chow [4].
///
same steps of the proof of Theorem 2.
I IXnkll
< oo, tightness is sufficient for Condition (iii) of
k,n
0 is also sufficient, tightness is not
However, since lim sup
Since sup
Theorem 2.
.k=iIankl
n
necessary.
Also, the more restrictive condition (iii) in Theorem 3 (than condition
(iii) of Theorem 2) allows the use of more general weights
3.
{ank}.
COMPLETE CONVERGENQE USING BASIS TECHNIQUES.
In this section the complete convergence of weighted sums will be established
for random elements in a Banach space which has a Schauder basis.
Since the con-
sistency of the kernel density estimates depend on a particular space which may have
a basis, more general results can be obtained.
The hypotheses will be shown to be
more applicable when the Baach space has a Schauder basis.
In particular, only
coordinate-wise independence for some Schauder basis will be needed, and the weights
{ank}
need only satisfy
exp[-/A
n
-n=l
2
> 0 where A
for each
(3.1)
<
n
k=l
ank"
Let {b.} denote a Schauder basis for E and let
{f.} denote
the corresponding
i
coordinate functionals.
Qt(x)
i;t+l
fi(x)bi
For notation convenience let U t (x)
for each t
1,2,3,...
.ti=l fi(x)bi
and
It can be assumed without loss of
generality that the basis is monotone (by renorming the space if necessary).
THEOREM 4.
basis.
If
Let
{Xnk}
be random elements in a Banach space which has a Schauder
R.L. TAYLOR
76
(i)
k > l} are independent, sub-Gaussian random
fi(Xnk):
for each i,
T(fi(Xnk))
variables for each n-with uniformly bounded parameters
(ii)
EXnk
(iii)
lira .up
t-o n
0 for each n and k, and
B
Q
ank Xnk) ll
t(.k=l
0,
then
k=l ank XnII
PROOF.
0
completely.
Given g > 0 pick t so that
IZk= I ank Qt(Xnk) II
sup
<
(3.2)
2
Thus, for each n
with probability one.
,k=l
-<
Xnkll
ank
> e]
P[llut (Zk:l ank Xnk) II
<.t
k=l
i=l
>
ank fi O%nk l
>
(3.3)
2t
nce, from (i), (ii), (3.1) ,and (3.3)
///
Theorem 4 allows for more general weights since
Note that in this case
bounded.
Zk=l ank Xnk
{fi(Xnk)}
(fi(Xnk
lim sup
t
n,k
<-
21/211fill
I IQ(Xnk) II
case when
Zk=l lankl
(supl Ixl I).
Also, by
bn
need not be
is assumed to be a.s. convergent.
For example, if
dition (i) is easily satisfied in general.
and k where K is a compact set, then
Zk:l lankl
P[Xnk
E
K]
Con-
i for each n
are sub-Gaussian with parameter
eemma
1.3.3 of Taylor [3]
xK
0 when
< i.
P[Xnk
E
K]
i.
Thus, Condition (iii) also holds in this
Finally, as an aside, the conclusion of Theorem 4 hold.
almost surely (rather than completely) if the uniform almost sure convergence of
Condition (iii) is replaced by simply almost sure convergence.
Coordinate-wise sub-Gaussian random elements are also appealing for applications
since coordinate-wise independence suffices in this setting.
In general, condition
COMPLETE CONVERGENCE FOR WEIGHTED SUMS
(iii) is the troublesome requirement which must be satisfied.
(a modification of Lemma 8.1 from Zaanen [6])allows (iii)
77
The following
lemma
to be replaced by a more
restrictive condition but one which may be easier to verify in applications.
Let {Y
LEMMA.
Ytn
(ii)
for fixed
(iii) Y
be real-valued random variables satisfying
-> 0 with probability one for all
(i)
n,Y(t+l)n -< Ytn
I, 2,
t
i, 2
t,n
tn
t and n,
with probability one for each
and
Y with probability one for each t and n where Y is a random
tn
variable such that EY < m.
Then, lira sup EY
n
t
THEOREM 5.
tone basis.
0 implies that lira E[sup Y
tn
n
t
tn
Let
{Xnk}
0.
be random elements in a Banach space which has a mono-
If
(same as Condition (i) and (ii) of Theorem 4),
(i) & (ii)
(iii)
lira sup
t
n
(iv)
E[sup
n
E[llQt(k=l ank Xnk) ll]
0, and
lk=l ank Xnkll] < o.
Then,
lk=l ank Xnkll
PROOF.
one.
ability one.
Thus,
lira EY
Hence
t
0.
k=l. ank Xnk
Ik=l ank Xnkll
is defined with probability
can be identified as Y in the pre-
Ytn I]Qt(.k=l ank Xnk) II satisfies
Next let Y be identified as sup If0 t(k=l
ank Xnk) II with probt
n
(iii) implies that lira E[sup If0 t(k=l
ank Xnk) II]
n
t-o
Since E has a monotone basis,
(ii) of the lemma.
t-o
a.s.
Condition (iv) implies that
With probability one, sup
n
ceding lemma.
0
lira
t-o
Yt
0 in probability, and lira
Yt m
0 with probability one
R.L. TAYLOR
78
{tm}"
for some subsequence
Since 0
Yt
-<
I0t(Ok=l an1. Xnk)[I
sup
n
ncreasing with probability one, it follows that lim
t-
-
lira sup
n
t
I[Q t(.k= Iank
Xnk)
is non-
Yt
Theorem 5 now follows from
0 with probability one.
III
a modification of Theorem 4.
The comparisons of general Banach space results with the results of this
section will be completed by showing that condition (iv) of Theorem 3 implies
condition (ii) of Theorem 5.
.k=iIankl
Assume that
<- I for each n and that for
compact, convex, symmetric set K can be chosen so that 0 e K and
sup
n
k=l lankl
EFllXnklll[Xnk
By Lemma 1.3.3 of Taylor [3], choose
sup
xGK
llQt(x)ll
<
.
Then for t >- t
t
k=l lankl
-< i for each n.
sup
n
!2.
(3.4)
so that for all t e t
O
o
K] )If
IQt(/.k=l ank Xnk I [Xnk
since
<
K]
<
Then for t -> t
O
-
O
(3 5)
2
E[IIQt (OO,k=l ank Xnk)
sup E[
n
+
sup
n
IQ t(.k=l ank
’k=l ankl
E[I
Xnk I[Xnk e K])I I]
Q
t
(Xnk) II l[Xnk
K]
<+=
from (3.4) and (3.5) since
liQt(x) II
<-
If
x
If
for all t.
Thus, the compact uniform
integrability of (3.4) (and hence tightness with uniformly bounded
pth
(p > 1)
moments) is sufficient for Condition (iii) of Theorem 5.
Identical distributions
and moments conditions will suffice for Condition (iii).
However, while it is often
true (in applications) that
{Xnk:
k >_ i} are independent and identically distributed
for each n, identical distributions seldom hold for the array
{Xnk}.
COMPLETE CONVERGENCE FOR WEIGHTED SUMS
ACKNOWLEDGEMENTS.
The research was supported in part by the Air Force Office of Scientific
Research under Contract Number F49620-79-C-140 and Grant Number AFOSR-81-0166.
Helpful editorial changes and suggestions are also gratefully acknowledged.
REFERENCES
[I]
WEI, D. and TAYLOR, R.L.
Convergence of weighted sums of tight
random elements, J. Multivariate Anal., 8, (1978), 282-294.
[2]
HOWELL, J. O. & TAYLOR, R. L.
[3j
TAYLOR, R. L.
[4]
CHOW, Y. S.
[5]
WOYCZYNSKI, W.
[611
ZAANEN, A.C.
Marcinkiewicz-Zygmund weak laws of
large numbers for unconditional random elements in Banach spaces.
Probability in Banach Spaces III LECTURES NOTES IN MATHEMATICS
Vol 860, (1981), Springer-Verlag, 219-230.
Stochastic Convergence of Weighted Sums of Random
Elements in Linear Spaces, Springer-Verlag Lecture Notes,
Vol. 672, (1978).
Some convergence theorems for independent random
variables, Annals Math. Statist., 37 (1966), 1482-1493.
On Marcinkiewicz-Zygmund laws of large numbers in
Banach spaces and related rates of convergence, Prob. and Math.
Statist., i, (1980), 117-132.
The Radon-Nikodym theorem
(1961), 171-187.
II, Inda ._th., __23,
79