Internat. J. Math. & Math. Sci.
Vol. 8 No. 4 (1985) 805-812
805
A NOTE ON CONVERGENCE OF WEIGHTED SUMS
OF RANDOM VARIABLES
XIANG CHENWANG
Department of Mathematics
Jilin University
Changchun
China
M. BHASKARA RAO
Department of Probability &
Statistics, The University,
Sheffield $3 7RH, U.K.
and University of Pittsburgh
(Received January 3, 1984, and in revised form July 25, 1984)
ABSTRACT.
Under uniform integrability condition, some Weak Laws of large numbers are
established for weighted sums of random variables generalizing results of Rohatgi,
Pruitt and Khintchine.
Some Strong Laws of Large Numbers are proved for weighted sums
of pairwise independent random variables generalizing results of Jamison, Orey and
Pruitt and Etemadi.
KEY WORDS AND PHRASES. Independent and pairwise independent random variables, weighted
sums of random variables, uniform integrability, convergence in probability, in mean,
strong convergence, random elements in separable Banach spaces.
1980 AMS SUBJECT CLASSIFICATION CODE. 60F05, 60F15.
be a sequence of real random variables defined on a probability
Let Xn, n
a double array of real numbers. Limit theorems
space (, B, P) and ank, n _> I, k _>
have been studied in the literature for the sequence Z ankX k, n _>
of weighted sums
k>l
of the sequence Xn, n _>
under some conditions on the double array of numbers and on
the distribution of the sequence Xn, n
I. Jamison, Orey and Pruitt [4] studied
almost sure convergence of weighted sums under the assumption that the sequence Xn, n
is independently identically distributed with
EIXII
=.
_>
One of the objects of this
paper is to extend the result of Jamison, Orey and Pruitt [4] on almost sure convergence
to cover the case of pairwise independent identically distrubuted sequences Xn, n
with
EIXII
EIXII
<
=.
Recently, Etemadi [3] has shown that Strong Law of Large Numbers is
.
valid for sequences Xn, n
which are pairwise independent identically distributed
<
with
The main result of Section 3 covers Etemadi’s result.
Our second objective in this paper is to study convergence in probability of the
sequence of weighted sums described above.
Convergence in probability has been studied
under the following condition.
(B)
There is a random variable X on [ such that
P{lXnl t
x}
! P{IXI
x}
EIXI s
<
for some s
t
0 and
806
X.C. WANG AND M.B. RAO
0 and n
for every x
See Rohatgi [7].
i.
Wei and Taylor
[9, Lemma 3, p.284] have
shown that if
(A)
EIXn lr
Su
<
for some r > 0
n>
holds, then-(B) holds for every 0
s
<r.
In this paper, we study convergence in
probability for sequences of weighted sums under the condition that
is uniformly integrable.
(C) Xn, n !
One can show that if (B) holds, then
IXn Is,
n
is uniformly integrable.
Chung [2, Exercise 7, 4.5].
One can also give examples of sequences X n, n
ing (C) but not (B) with s
I.
2.
!
See
satisfy-
CONVERGENCE IN PROBABILITY.
In this section, we present some results on convergence of weighted sums from which
Weak Law of Large Numbers is derivable.
Xn,
Xn,
Let
THEOREM i.
such that
n
_>
Theorem
generalizes some results in the
See the remarks following Theorem i.
literature in this area.
be a sequence of pairwise independent random variables
n
Let ank, n
is uniformly integrable.
_> I,
k
_>
be a double array of
real numbers satisfying
lankl
C
lankl,
n
Z
(i)
k>l
(ii) max
for some constant C > 0 and
for every n
converges to O.
J
k>l
Z
Then
k>l
ank(X k
EX k), n
converges to 0 in the mean.
Z
It is clear that the series
PROOF.
ank(X k
k>l
[m] for every n
!
since sup
k>l
that lim
P{I
Z
ank(X k
k>l
n/
lXkl
whenever A
k
g
<
Z
and
k>l
t}
EXk)
tegrable, there exists
sup
k>l
mlXkl
0.
Let s
O.
is convergent.
Since
X,
k
Let t>O.
We show
is uniformly in-
0 such that
dP < gt/8C
and P(A) < 6.
(2.1)
Further, by Chebychev’s inequality, for any m
0 and
i,
P{IXkl
(I/m)mlXkl
>m}
(I/m) sup
k>l
Consequently, there exists a
sup
k>1
e{IXkl
Yk
Zk
that Yk’
6.
Xk
if
IXkl
0
otherwise, and
IYk EYkl
Xk
k
ElXkl.
0 such that
a}
Define for every k
Note
lank
EX k) converges absolutely a.e.
(2.2)
I,
a,
Yk"
is a sequence of pairwise independent random’variables satisfying
2a for every k
EIZkl "{IXkl/ a
! I.
IXkl
By (2.1) and (2.2), we have for every k
dP <
et/8C.
,
CONVERGENCE OF WEIGHTED SUMS OF RANDOM VARIABLES
i,
Consequently, for every n
I
E
Therefore, by
EZk) _<
ank(Z k-
k>
807
lankl EIZ k -EZkl
I
k>
lankl ElZkl
E
2
k>
t/4.
Chebychev’s inequality, for every n _> I,
P{I
E
(2.3)
/2.
t/2}
ank(Zk- EZk)
k>l
t2/B2a2C.
maxlank
k>l
N,
We observe that for every n
P{I
E
t/2}
ank(Y k -EYk)
k>l
N, we have
such that for every n
Next, we choose N
_< (4/t 2) Var(
Y.
ank(Yk- EYk))
k>l
(4/t 2)
E
k>
EYk)2
2
ank E (Yk
(4/t2)(max lankl)
E
k>
k
lanklE(Y k EYe )2
(2.4)
e/2
Finally, (2.3) and (2.4) yield
P{l
t}
EXk)
ank(Xk
_< P{l
E
ank(Y k
k>
k>
+
P{I
E
k>l
< e
Thus
E
k>l
EXk),
ank(X k
J
n
EZ k)
ank(Z k
N.
for every n
E
k>l
integrable and
E
k>
t/2}
converges to 0 in probability.
To establish mean convergence, it suffices to show that
uniformly integrable.
>t/2}
EYk)
ank(X k
EXk),
n
is
is uniformly
Since X k, k
I, it is obvious that E ank(Xk
EXk), n
k>
See Chung [2, Theorem 4.5.4, p.97].
lankl
!
C for every n
is uniformly integrable.
REMARKS.
(I).
Rohatgi [7, Theorem i, p. 305] showed that
E
k>1
ank(Xk
EXk),
n
converges to 0 in probability under the following conditions.
(i)
(ii)
(iii)
Xn, n
is independent.
(B) holds with s
i.
The double array ank, n
!
i, k
of real numbers satisfies (i) and (ii) of
Theorem i.
In view of the remarks made in the introduction, Theorem
Rohatgi.
generalizes this result of
(Also, this result of Rohatgi was a generalization of a result of Pruitt
is
[6, Theorem i, p. 770] who started with the assumption that the sequence Xn, n !
independently identically distributed with
). Moreover, our proof is simpler
EIXII
than the one presented by Rohatgi.
The essential difference in the proofs lies in the
fact that we truncate each X n at a fixed point a, where as Rohatgi truncated X n at a n
of Rohatgi,
over Theorem
with a n varying with n. To illustrate the power of Theorem
a
of
be
Let X n, n _>
consider the following example.
sequence
pairwise independent
random variable with X n having the following probability law.
X.C. WANG AND M.B. RAO
808
_>
Xn, n
P{Xn
n}
P{Xn
0}
P{Xn
-n}
I/2nlog(n+l),
(I/nlog(n+l)).
Rohatgi’s theorem is not applicable to determine the convergence of
n
E
(i/n)
n
But by Theorem I, the sequence (I/n)
to 0 in probability.
X k, n
k=l
n >
_>
But (B) does not hold for the sequence Xn, n
is uniformly integrable.
i.
with s
E
Xk,
k=l
does indeed converge to 0 in probability.
Chung [2, Theorem 5.2.2, p. 109] proved (attributed to Khintchine) the result
(2).
+ X2 +
that (i/n)(X
+ Xn),
n
!
converges to EX
in probability if Xn, n
is
a sequence of pairwise independent identically distributed random variables with
EIXII
.
<
generalizes this result, Moreover, the proof presented here is
Theorem
much simpler than the one presented by Chung.
If we impose a stronger condition on the double array, we can establish a Weak Law
of Large Numbers for weighted sums without the assumption of independence of the random
variables but in the presence of uniform integrability.
_>
Let X n, n
THEOREM 2.
be a sequence of real random variables defined on a
probability space (,,P) such that
0 <r <I.
_>
Let ank, n
lank Ir
E
(i)
k >I
(ii)
_>
i, k
lankl,
max
IXn Ir,
n
!
is uniformly integrable for some
be a double array of real numbers satisfying
C for every n
O,
for some constant C
converges to 0.
n
k>l
E
k>l
Then
anuXu,
n
E
The series
ianklrlXk Ir
_>
n
ankXk
k>l
E
converges to 0 in r-th mean.
This can be proved by a simple modification of the proof of Theorem I.
PROOF.
k>l
_>
converges absolutely a.e.
converges a.e. [P] and 0 < r < I.
sup /
k>l A
IXk Ir
{P]
_>
for every n
The inequality (2.1) takes the form
et/8C,
dP
and the inequality (2.2) remains intact as it is.
The sequences
Yk’
k
are defined in exactly the same way as it was done in the above proof.
P{I
E
k>l
I the
ankZkl
t/2} is estimated by
E
k>l
lank Ir EIZk Ir.
E
proof of Theorem I, we showed that
k>l
in probability.
sequence
E
k>l
ank(Y k
!
and Z k, k
The probability
Now comes the point of departure.
EYk),
converges to zero
k
Under the conditions of Theorem 2, we can do better than this.
ank Yk’
does indeed converge to 0 a.e. [P].
n
For every n
following chain of inequalities.
k>
since
a
ankYk
It now follows that
E
k>
E
k>l
ankXk
lankl
n
!
a(max
k>
_>
The
This follows from the
i,
lankl) l-r
E
k>
lank Ir
converges to 0 in probability.
To establish con-
809
CONVERGINCE OF WEIGHTED SUMS OF RANDOM VARIABLES
Z
vergence in r-th mean, it suffices to show that
k>l
tegrable.
ankXk Ir,
is uniformly in-
n
This is not hard to prove.
Rohatgi [7, Theorem i, p.305] established a weaker conslusion than the
REMARKS.
one given above under stronger conditions that the sequence X n, n
n _>
distributed and that (B) holds for the sequence
with s
Xn,
is independently
The major im-
r.
provement achieved by Theorem 2 over Rohatgi’s theorem is dropping the assumption of
independence.
Now, we enquire about the validity of Theorems
and 2 in the context of separable
Theorem 2 is valid for sequences of random variables taking values in
Banach spaces.
For the sake of clarity, we give the statement below.
separable Banach spaces.
Let X n, n _>
be a sequence of random elements taking values in a
separable Banach space B equipped with a norm
such that
n
is uniTHEOREM 3.
llXnll r,
II-II
Let ank, n _> i, k _>
be a double array of real
numbers satisfying (i) and (ii) of Theorem 2. Then Z ankX k, n
converges to 0 in
k>l
formly integrable for some 0 <r <I.
the r-th mean, i.e.,
REMARKS.
2.
(I).
Eli
k>l
ankXkll r,
n
converges to zero.
The proof of Theorem 3 is analogous to the one given for Theorem
This result is a generalization of Theorem 2.1 of [I].
The major improvement
achieved in the above result is in disposing of the assumption of independence in
Theorem 2.1 of [I].
Moreover, the proof suggested above is much simpler than the one
presented in [I] for Theorem 2.1.
(2).
Theorem
is not valid for Banach space-valued random variables under con-
ditions similar to those imposed in Theorem i.
of [I]).
(See the comments following Theorem I.I
For the validity of Theorem 3, almost sure convergence of
Z
k>l
to 0 certainly helped.
The proof given for Theorem
n
fails to work in Banach spaces
Z
because we are unable to establish convergence of
ankY k,
k>l
ankY k,
n
_>
to 0 either in
probability or a.e. [P].
However, if X n, n
is uniformly tight, i.e., given e
O,
there exists a compact subset C of B such that P{X
C}
e for every n _> I, then
n
Theorem
is valid under the conditions stipulated therein.
For further details on this
result, see Wang and Bhaskara Rao [i0, Theorem 2.4].
3.
oN-STRONG
CONVERGENCE.
Extensions of
Kolmogorov’s Strong Law of Large Numbers are generally sought
so that
they become more applicable under circumstances less stringent than those imposed by
Jamison, Orey and Pruitt [4, Theorem 3, p. 42]
Kolmogorov’s Strong Law of Large Numbers.
worked with independent identically distributed sequences of random variables but im-
posed conditions on the weights to establish Strong Law of Large Numbers.
Etemadi
[3, Theorem i, p. 119] relaxed the assumption of independence in Kolmogorov’s Strong
Law of Large Numbers to pairwise independence and arrived at the same conclusion.
The
following result encompasses both these extensions.
.
be a sequence of pairwise independent identically disTHEOREM 4. Let Xn, n
tributed random variables defined on a probability space (,8,P) satisfying
EIXII
Let a n
n
be a sequence of positive numbers satisfying lim max
n
l<i<n
(ai/An)
0, where
X.C. WANG AND M.B. RAO
810
n
An
l ai, n
i=l
(Ak/a k)
_>
I.
If N(n)/n < r for every n >
O.
<n, n
n
Z
Let N(n) be the number of positive integers k
(ak/An)X k,
_>
n
_>
such that
0, then
for some constant r
a.e.[P].
converges to EX
k=l
The proof is carried out in the following three major steps.
PROOF.
Since
and
satisfies the hypothesis of the theorem,
n _>
each of the sequences X+
n, n _>
For each k _> I, let
we can assume, without loss of generality, that each Xn _>0"
X,
Ak/a k,
if X k <
otherwise.
Xk
Yk
0
Note that
shown that
Yk’
Z
k
_>
(a
n>l
It can be
is a sequence of pairwise independent random variables.
n
Further, I
n _>
is convergent.
converges to
n2/A2n )EYn2
(ak/An)Xk,
k=l
n
converges to 0 a.e. [P].
Z (ak/An)(Y k
EYk), n
k=l
details are worked out in Stout [8, p. 221].
EX
a.e.[P] if and only if
The
We now show that almost sure convergence takes place along some well chosen
2
n
Z (ak/An)(Y k
EYk), n I. In the final step 3
k=l
show that convergence takes place along the entire sequence almost surely.
special subsequences of
of positive integers by letting m
Define a sequence m I, m 2
a}, i
{j _> i; Aj _>
Z
k-I
(ak/Ami )(Yk
EYk)’
O,
any e
2,3
Ami_l
mi
mi
i>!Z P{Ik.iI
Ba,e
converges to 0 a.e.[P].
i
m
i
mi
Z
Z
i>l k=l
(I/e 2)
Z
k>
where
It is clear that
(i/Am2j)(a2/a2
(ak/Am)(y
kl -EYk))
i>iZ k=lZ (a/)i" War
<_ (I/e2)
a my2
i)
i/ 2
Z
k
> A and
k
Amj k-
_<
(Yk)
(a2k/)i EY
>J k
min{j >_ I; mj
Jk
It suffices to show that for
mi
(I/e2)i>IZ Var(k=iZ
(I/2)
I.
Let a
min
and m i
We show that
} < ="
>
EYk)
(ak/Am i)(Yk
By Chebychev’s inequality,
Ba’ -<
<m 2 <m 3 <
Obviously, m
we will
l
Aj,
>_k},
i/
J>-Jk
<
(I/
1,2,3
6
)(i + I/a 2 + I/a 4 + i/a +
Jk
(i/A)(a2/a2-1).
k
Consequently,
k>l
n
3
to 0
Finally we show that the entire sequence
a.e.[P].
Since lira max
<k<n
n
(ak/An)
O, lira
n+
I (ak/An)(Yk
k=l
(an/An_
EYk)
0 and lira
n+
n
(An/An_
converges
I.
CONVERGENCE OF WEIGHTED SUMS OF RANDOM VARIABLES
Observe also that
<
n
(An/An_l)
<
An/An_
for every n
n_>N.
for all
Ami < Ami-I
2/Ami
Consequently, I/An <
E
k=l
(ak/An)Y k
Therefore,
n
lim sup
n/
(ak/An)Y k _<
E
k=l
(One can check that lim EY n
n+
EX
mi
lim I
i
k=l
(ak/Ami-i)
_> N a.
if mi_
r.
n+
k=l
k=
n/
2
E
k=l
(ak/Ami)Y k 2EXI
I/2Ami_l
n
and
or
(ak/Ami)Yk
then
E
k=l
every n
_>
N.
[P].
a.e.
k=
mi-1
r.
i/
k=l
EXI.
(ak/An)EY k
(ak/An) Yk-> (I/2)
mi-I
y.
k=l
(ak/ami_) Yk (1/a2)EXl
a.e.
[PI-
I,
(ak/An)Y k
< lim inf
N,
If mi_
(3.I)
mi
n/
lim
Thus we observe that for every
(I/2)EX
For each
Consequently,
tl/a2)
(ak/An) Yk
An_ I.
n < m i.
n
I
n
lim inf
< A <
n
and, by Toeplitz lemma, lim
From (3.1), we also observe that I/An
Yk’
n>N, An_
such that
2An.
n
and
N
We can find an integer
2 such that mi_
2Am
<
2.
Equivalently, if
I, there exists a unique integer i
An <
_>
811
< lim sup
n/
n
I
k=
(ak/An)Y k _< 2EX
a.e. [e].
This proves
the almost sure convergence of the desired sequence.
REMARKS.
(i).
This theorem extends to separable Banachspaces-valued random
A proof can easily be obtained with appropriate modifications of
variables verbatim.
the proof of Strong Law of Large Numbers given in Padgett and Taylor [5, p.42-44].
one can adopt the argument given in Bozorgnia and Bhaskara Rao
Or,
[I] in the proof of
their Theorem 2.1.
Kolmogorov’s inequality plays a crucial role in the standard proof offered
in many te.t books for Kolmogorov’s Strong Law of Large Numbers.
This inequality, as
(2).
it stands, cannot be commandeered for pairwise independent sequences of random variables.
The idea of establishing convergence along some special subsequences is taken from
Etemadi
[3] but his technique has been modified extensively in the above proof
to suit
Also, Theorem 4 strengthens the conclusion of Theorem 5.2.2 of Chung [2,
our needs.
p. 109] from convergence in probability to convergence almost everywhere [P].
There are sequences a n
but N(n)/n, n
of positive numbers such that lim max
n+ <k<n
n
is unbounded.
Theorem 4 becomes inapplicable.
,
EIX 111og+Ixl
See Jamison, Orey and Pruitt [4, p.43].
ak/An
0
In such a case,
However, if we impose a stronger condition that
one can establish a Strong Law of Large Numbers generalizing Theorem
4 of Jamison, Orey and Pruitt [4, p.43] as follows.
THEOREM 5.
Let Xn, n
_>
be a sequence of pairwise independent identically dis-
tributed real random variables defined on a probability space (,B,P) with
LIE lllog+IXl
=.
n
where An
E
a i, n
Let a n
_> I.
n
_>
be a sequence of positive numbers satisfying lim
Then
i=l
n
E
k=l
(ak/An) Xk,
n
_>
converges to EX
a.e.
[P].
An=
,
X.C. WANG AND M.B. RAO
812
PROOF.
Using Lemma 2 of Jamison, Orey and Pruitt [4, p.43], one can prove the above
result by a suitable modification of the proof of Theorem 4.
REMARK.
Theorem 5 is also valid for separable Banach space-valued random variables.
The relevant moment condition is that
ACKNOWLEDGEMENTS.
EllXllllog+llXlll
<
The authors are grateful to the referee for his comments and his
suggestions to improve the tone of the paper.
Part of the work of the second author was supported by the Air Force Office of
Scientific Research under Contract F49620-82-0001.
Reproduction in whole or in part
is permitted for any purpose of the United States Government.
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BOZORGNIA, A. and BHASKARA RAO, M. Limit theorems for Weighted Sums of Random
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CHUNG, K.L. A Course in Probability Theory, Second Edition, Academic Press, New
York, i$.
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ETEMADI, N.
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PADGETT, W. and TAYLOR, R.L. Laws of .Large Numbers for_N0rmed
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Berlin, 1973.
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PRUITT, W.E.
An Elementary Proof of the Strong Law of Large Numbers, Z.
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Summability of Independent Random variables, J. Math. Mech., 15 (1966),
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7.
ROHATGI, V.K.
8.
STOUT, W.F.
9.
WEI, W. and TAYLOR, R.L. Convergence of Weighted Sums of Tight Random Elements,
J. Multivariate Anal., 8 (1978), 282-294.
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WANG, X.Co and BHASKARA RAO, M. Convergence in the p-th mean and Some Weak Laws
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conversence,
Academic Press, New York, 1974.