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TAIWANESE JOURNAL OF MATHEMATICSVol. 15, No. 4, pp. 1759-1781, August 2011
This paper is available online at http://www.tjm.nsysu.edu.tw/
ALMOST SURE AND COMPLETE CONVERGENCE OF RANDOMLY
WEIGHTED SUMS OF INDEPENDENT RANDOM ELEMENTS IN
BANACH SPACES
Le Van Thanh1 and G. Yin2
Abstract. This work develops almost sure and complete convergence of
randomly weighted sums of independent random elements in a real separableBanach space. Sufficient conditions for almost sure convergence and complete
convergence in the sense defined by Hsu and Robbins are provided. Examples
showing that the conditions cannot be removed or weakened are given. It is
also demonstrated that some of the known theorems in the literature are special
cases of our results.
1. INTRODUCTION
According to Hsu and Robbins [9], a sequence of real-valued random variables
{Xn, n 1} converges completely to 0 if,
(1.1)
n=1
P
|Xn| >
< for all > 0.
Based on this definition, Hsu and Robbins [9] and Erdos [7] derived a necessary
and sufficient condition for a sequence of independent identically distributed (i.i.d.)
random variables{Xn, n 1}. The assertion is: EX1 = 0 andE|X1|2 < if
and only if
(1.2)
n=1
P n
i=1
Xi> n< for all >0.
Received April 12, 2010, accepted April 12, 2010.
Communicated by Jen-Chih Yao.2000 Mathematics Subject Classification: 60F15, 60B11, 60B12.
Key words and phrases: Almost sure convergence, Complete convergence, Random weight, Row-wise
independent, Banach space.1Research of this author was supported in part by the National Science Foundation under DMS-
0603287.2 Research of this author was supported in part by the National Science Foundation under DMS-
0907753, and in part by the Air Force Office of Scientific Research under FA9550-10-1-0210.
1759
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1760 Le Van Thanh and G. Yin
Their theorem was extended by Baum and Katz [4] to establish a rate of convergence
in the sense of Marcinkiewicz-Zygmunds strong law of large numbers, which reads:
Let > 1/2, p 1, and{Xn, n 1} be a sequence of i.i.d. random variables.ThenE X1= 0 andE|X1|
p < if and only if
(1.3)
n=1
np2P n
i=1
Xi> n< for all > 0.
The results have been generalized and extended in several directions. A host of
researchers considered the problem in a Banach space setting; see Ahmed et al. [3],
Hu et al. [11], Sung [21], Hernandez et al. [8], Sung et al. [19, 22], and Wang[25] among others.
Recall that a sequence of random elements {Vni, n 1, 1 i n} is stochas-tically dominatedby a random elementV if for some constant C < ,
(1.4) P{Vni > t} CP{V > t}, t 0, n 1, 1 i n.
Using stochastic dominance, Hu et al. [10] treated triangular arrays of row-wise
independent random variables and obtained the following complete convergence
result with a Marcinkiewicz-Zygmund type normalization. Let{Xni, n 1, 1 i n} be an array of row-wise independent mean 0 random variables that isstochastically dominated by a random variableX, that is, condition (1.4) is satisfied.
IfE|X|2t < where1 t n1/t< for all > 0.
Subsequently, Wang et al. [25] considered triangular arrays of row-wise independent
random elements and obtained the following convergence rate. Let1/2 < 1and{Vni, n 1, 1 i n} be an array of row-wise independent random elements,which is stochastically dominated by a random elementVin the sense of condition(1.4) being satisfied. IfEVp < for some p >1 and
(1.6) maxkn
Pk
i=1 Vni> n 0 as n for all >0,
then
(1.7)
n=1
np2P n
i=1
Vni> n< for all >0.
Later, Sung established a complete convergence for weighted sums of random ele-
ments in [20]. Let{Vni, n 1, 1 i n} be an array of row-wise independentrandom elements that is stochastically dominated by a random element V. Letp 1and{bni, n 1, 1 i n} be an array of constants satisfying
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Randomly Weighted Sums in Banach Spaces 1761
(1.8) max1in
|bni| =O(n1/p)
and
(1.9)
ni=1
b2ni= o 1
log n
.
IfEV2p < and
(1.10)
n
i=1bniVni 0 in probability asn ,
then
(1.11)
n=1
P n
i=1
bniVni> < for all >0.
A number of researchers considered the weighted sums of random elements
where the weights are constants; see Ahmed et al. [3], Hu et al. [11], and Sung
[21]. In [16], Li et al. improved the main result of Thrum [24]; their result
reads: Let {Vi, i 1} be a sequence of independent identically distributed randomelements with EV1
p < for some p 1. Let {ani, n 1, 1 i n} be an
array of constants such that supn,i |ani|< and
(1.12)
ni=1
a2ni= O(n) for some 0 < < min{1, 2/p}.
If
(1.13) 1
n1/p
ni=1
aniVi 0 in probability,
then
(1.14) 1
n1/p
n
i=1 aniVi 0 a.s.In this paper, we obtain the almost sure convergence and complete convergence
for randomly weighted sums of arrays of row-wise independent random elements.
Although Cuzick [5], Li and Tomkins [18] and Li et al. [17] also considered almost
sure convergence with random weights, to the best of our knowledge, there has yet
been any result on the complete convergence of the case in which the weights are
random variables. Moreover, the results in the aforementioned papers do not imply
our results even in the case the weights are non-random. Most of the results of the
aforementioned work in the literature can be covered and improved by our results.
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1762 Le Van Thanh and G. Yin
As a motivation, we describe a problem arising in systems theory. Recall that
a transfer function is commonly used in the analysis of single-input single-output
systems, in signal processing, communication theory, and control theory. Consider a
continuous-time, linear, time-invariant system. Then a transfer function is the ratio
of the Laplace transforms of the output and input. It gives input/output character-
istics in the frequency domain. In certain applications, the underlying systems need
to be identified, which can be carried out by estimating the transfer functions using
observable data. Suppose that we wish to identify an unknown transfer function
K(). The input is given by z(), sampling interval is , and output (at sampling
time n) isyn, where
yn =
10
K(u)z(n u)du + n,
where {n} is a sequence of observation noise, which is a sequence of i.i.d. randomvariables with 0 mean and finite variance and is independent of z(). To identifyor to estimate K(), we treat this as a stochastic optimization problem in L2[0, 1],which is the Hilbert space consisting of square integral functions defined on the
interval [0, 1]. Let
x, y
and ||x||= [
x, x
]1/2 denote the inner product and thenorm onL2[0, 1], respectively. To carry out the desired task, we construct algorithm
of the form
(1.15) Kn+1(t) =Kn(t) +nz(nt)yn 10Kn(u)z(n u)du.
The above algorithm is stochastic approximation type analogs to its finite dimen-
sional counterpart [14]. Define Kn(t) =Kn(t) K(t), and useKn, Kn, K,andzn denote the random variables whose point values areK(t), Kn(t), K(t),andzn(n t), resp.
(1.16) Kn+1(t) =Kn(t) + nzn[Kn, zn +n].In the implementation, one often wishes to use a projection algorithm
(1.17)
Kn+1(t) =
Kn(t) + nzn[
Kn, zn
+ n]
,
where () is the projection to the unit ball in L2[0, 1]. As explained in [13, p.782], this can be further rewritten as
(1.18) Kn+1(t) =Kn(t) +nzn[Kn, zn +n] ncnKn+On,wherecn satisfiesncn< 1, andOn satisfies
m(T)j=0 Oj 0, for any 0 < T <
with m(t) = max{n : tn t} and tn =n1
j=0j . In analyzing such algorithms,one often needs to deal with a term (see [13, p. 783])
nk=0
nj=k+1
(1 j cj)
kzk(
zk, Kk+ k).
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Randomly Weighted Sums in Banach Spaces 1763
As can be seen that this term is of the form a weighted sum of random variables.
Our work in this paper provides a sufficient condition for almost sure convergence
and complete convergence of such a sequence.
We also note that many linear models in statistics based on a random sample in-
volve weighted sums of dependent random variables. Examples include least-squares
estimators, nonparametric regression function estimators, and jackknife estimates,
among others. In this respect, the study of convergence for these weighted sums
has impact on statistics, probability, and their applications.
The rest of the paper is arranged as follows. Section 2 presents some definitions
and lemmas to be used in the proof of our results. Section 3 provided conditions foralmost sure convergence for randomly weighted sums. Our results in this section
improve the results by Li et al. [16]. Section 4 establishes complete convergence
for randomly weighted sums of arrays of row-wise independent random elements in
a real separable Banach space. In the main results, Theorem 3.1, Theorem 3.3, and
Theorem 4.1, no assumptions are made concerning the geometry of the underlying
Banach space. Finally, in Section 5, some corollaries and examples are presented
to illustrate the results. It demonstrates that our conditions are sharp and our results
lead to better convergence rates. Throughout this paper, the symbol C denotes a
generic positive constant (0< C < ) whose values may be different for differentappearances.
2. PRELIMINARIES
We begin with definitions, notation, and preliminary results prior to establishing
the main results. Let X be a real separable Banach space with norm ||||. A randomelement inXwill be denoted by V or by Vn, Wn, etc.
The expected value or mean of a random element V, denotedE V, is definedto be the Pettis integralprovided it exists. That is, Vhas expected value E V Xifh(EV) = E(h(V)) for every h X, whereX denotes the (dual) space of allcontinuous linear functionals on X. IfE||V||< , then (see, e.g., Taylor [23, p.40])Vhas an expected value. But the expected value can exist whenE||V|| = .
For an example, see Taylor [23, p. 41].Lemma 2.1 below is a Marcinkiewicz type inequality for some of independent
random elements; see de Acosta [1] for a proof.
Lemma 2.1. Let{Vni, n 1, 1 i n} be an array of row-wise independentrandom elements. Then for every r 1, there is a positive constantCr dependingonly on r such that for alln 1,
(2.19) E n
i=1
Vni En
i=1
Vnir Cr n
i=1
EVnir, for1 r 2,
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1764 Le Van Thanh and G. Yin
and
(2.20)
E n
i=1
Vni En
i=1
Vnir
Cr
ni=1
EVni2r/2
+n
i=1
EVnir
, forr >2.
The following lemma is a Banach space version of Lemma 2.1 of Li at al. [16].
The proof is similar to that of Lemma 2.2 in Hu at al. [11].
Lemma 2.2. Letr 1, and{Vn, n 1} be a sequence of random elementsand{V
n, n 1} an independent copy of{Vn, n 1}. IfVn 0 in probability,then
(i) Vn 0 a.s. if and only ifVn V
n 0 a.s.
(ii)
n=1nr2P{Vn }< for every >0 if and only if
n=1
nr2P{Vn
V
n } < for every > 0.
The third lemma due to Hu at al. [11] concerning the relationship between
convergence in probability and mean convergence for sums of independent bounded
random elements.
Lemma 2.3. Let{Vni, n 1, 1 i n} be an array of row-wise independentsymmetric random elements and suppose there exists > 0 such thatVni a.s. for alln 1, 1 i n. If
ni=1 Vni 0 in probability as n , then
En
i=1 Vni 0 asn .
The following lemma is a result of Adler at al. [2] concerning the expectation
of the truncation random elements that are stochastically dominated by a random
elementV.
Lemma 2.4. Let{Vni, n 1, 1 i n} be an array of random elements thatis stochastically dominated by a random elementV . Then for all p >0
(2.21)
E
VnipI(Vni t)
CtpP(V > t)+CE
VpI(V t)
, t 0, n 1, 1 i n,
and
(2.22)
E
VniI(Vni> t)
CE
VI(V >t)
, t 0, n 1, 1 i n.
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Randomly Weighted Sums in Banach Spaces 1765
Finally, we present two exponential inequalities for independent Banach valued
random elements. See Sung [20, Lemma 1] for a proof of Lemma 2.5 (see also
[15]). The proof of Lemma 2.6 is also similar to that of Sung [20, Lemma 1].
Lemma 2.5. Let{Vi, 1 i n} be independent random elements such thatVi A a.s. for some constantA >0. Then for allt >0,
E
exp(tn
i=1
Vi)
exp
tEn
i=1
Vi + 2t2
ni=1
EVi2 exp(2tA)
.
Lemma 2.6. Let{Vi, 1 i n} be independent random elements such thatfor each1 i nVi Ai a.s. for some constantsA i> 0. Then for allt >0,
E
exp(tn
i=1
Vi)
exp
tEn
i=1
Vi + 4t2
ni=1
EVi2g(2tAi)
,
where g (x) =x2(ex 1 x).
3. ALMOSTSURECONVERGENCE
In this section, we establish the almost sure convergence of randomly weightedsums of independent random elements. The first theorem is a new result even when
Ani are constants. This theorem improves the cited result of Li et al. [16].
Theorem 3.1. Let p 1, and {Vi, i 1} be a sequence of independentrandom elements that are stochastically dominated by a random elementV with
EVp 2 and
(3.3)
ni=1
E(|Ani|2) = o(n2/p log1 n),
then
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1766 Le Van Thanh and G. Yin
(3.4) limn
1
n1/p
ni=1
AniVi= 0 a.s.
(2) If1 p 2 and
(3.5)
ni=1
E(|Ani|p) =O(n) for some0 < 0, we choose a positive integerN(to be specialized later), and
let
Xni =ViI(AniVi n1/p
log1
n),
Yni =ViI(n1/p log1 n < AniVi n
1/p
N),
and
Zni= ViI(AniVi > n1/p
N).
Proof of part(1) . For all t >0, by Lemma 2.5
(3.7)
P(n
i=1
AniXni> n1/p)
exp(tn1/p)E exp(tn
i=1
AniXni) (by the Markov inequality) exp(tn1/p)
exp
tEn
i=1
AniXni + 2t2
ni=1
EAniXni2 exp(2tn1/p log1 n)
.
Since EVp < , p >2, we have from (3.1) and (3.3) that
(3.8)
ni=1
EAniXni2 C
ni=1
E|Ani|2 =o(n2/p log1 n).
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Randomly Weighted Sums in Banach Spaces 1767
Putting t = 2n1/p log n
, it follows from (3.2), (3.7) and (3.8) that there exists
1< n1/p) C
1
nand for all large n.
Therefore
(3.10)
n=1
P(n
i=1
AniXni > n1/p)< .
Now, we use the method in [12]. From the definition ofYni,
(3.11)
P(n
i=1
AniYni > n1/p)
P(there are at leastN suchis that AniVi> n1/p log1 n)
1i1 n1/p log1 n
n
i=1P(AniVi > n
1/p log1 n)
N
n
i=1
EAniVip
n logp n
N
Kp2n
i=1
E|Ani|2EVi
p
n logp n
N
Cn
i=1
E|Ani|2
n logp n
N C
logp nn12/p
N.
Choose Nlarge enough such that (1 2/p)N >1, we get from (3.11) that
(3.12)
n=1
P(n
i=1
AniYni> n1/p)< .
Now, we note that EVp < implies
i=1 P(Vi > i1/p) < , so by the
Borel-Cantelli lemma,
(3.13)
ni=1
Zni= O(1) a.s.
It follows from (3.1) and (3.13) that
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Randomly Weighted Sums in Banach Spaces 1769
Putting t = 3n1/p log n
, it follows from (3.17) and the second part of (3.6)
that
(3.18) P(n
i=1
AniUi> n1/p) C
1
n2and for all large n.
Therefore
(3.19)
n=1P(
n
i=1AniUi > n
1/p)< .
By the Borel-Cantelli lemma,
(3.20) n1/pn
i=1
AniUi 0 a.s. as n .
By the argument similar to that of (3.13), we haven
i=1 Wiis bounded a.s. Thisimplies
(3.21) n1/pn
i=1
AniWi K
n1/p log n
ni=1
Wi 0 a.s. as n .
Conclusion (3.4) follows from (3.20) and (3.21).
Remark 3.2. We note:
(i) When Ani ani a.s., where {ani, n 1, 1 i n} is a bounded arrayof constants, then both condition (3.2) and the second part of (3.6) can be
replaced by
(3.22) n1/pn
i=1
aniVi 0 in probability asn .
The proof is standard (see Remark 4.3).
(ii) When p > 2, the condition (1.12) of Li et al. [16] is stronger than (3.3).So, Theorem 3.1 improves the cited result of Li et al. [16] even when the
non-randomly weighted case.
(iii) We now compare Part (1) of Theorem 3.1 with result of Sung [20] citedin Section (when p > 2). When Ani ani n
1/pbni, (3.1), (3.3) and
(3.22) coincide with (1.8), (1.9), (1.10), respectively. Sung [20] obtained the
complete convergence ofn
i=1 bniVni under condition EV2p < while
in Theorem 3.1, we obtain the almost sure convergencen
i=1 bniVi underconditionEVp < .
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1770 Le Van Thanh and G. Yin
(iv) Condition (3.5) is strong, but as we can see in Part (3), when (3.1) is strength-
ened tomax1in |Ani| K/ log n a.s., (3.5) can be removed. According toa result by Cuzick [5, Corollary 2.1 and Lemma 2.1], Part (3) does not hold
ifmax1in |Ani| K/ log n a.s. is weakened to (3.1).
The proof of the following theorem is similar to that of Part (1) of Theorem 3.1.
Theorem 3.3. Let p > 2, and {Vi, i 1} be a sequence of independentrandom elements that are stochastically dominated by a random elementV withEVp 1/2, 1 p < 2, {Vni, n 1, 1 i n} be an arrayof row-wise independent random elements that are stochastically dominated by a
random elementV with EVp < . Let{Ani, n 1, 1 i n} be an arrayof row-wise independent random variables. Assume that for all n 1, the randomvariables{Ani, 1 i n} are independent of{Vni,1 i n}. If
(4.1)
ni=1
E(|Ani|2) =O(n),
and
(4.2) limn
nE n
i=1
AniVniI(Vni n)= 0,
then
(4.3)
n=1
np2P n
i=1
AniVni> n< for all > 0.
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Randomly Weighted Sums in Banach Spaces 1771
Proof. Noting that (4.3) is automatic ifp 0,
(4.4)
P n
i=1
AniVni> n
P
max1in
Vni > n
+P
n
i=1
AniWni > n
.
To obtain (4.3), it remains to show that
(4.5)
n=1
np2P
max1in
Vni > n
< ,
and
(4.6)
n=1
np2P
n
i=1
AniWni > n
< .
First, we have
(4.7)
n=1
np2P max1in
Vni> n
n=1
np2n
i=1
P
Vni> n
C
n=1
np1P(V > n)
=C
n=1
np1
i=n
P(i < V (i + 1))
=C
i=1i
n=1np1P(i < V (i + 1))
C
i=1
ipP(i < V (i + 1))
CEVp < .
This proves (4.5). Note that it follows from (4.2) that for all largen,
(4.8) nE n
i=1
AniVniI(Vni n)< /2.
So, to prove (4.6), it suffice to show that
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1772 Le Van Thanh and G. Yin
(4.9)
n=1
np2P n
i=1
AniWni En
i=1
AniWni> n/2< .
We have
n=1
np2P n
i=1
AniWni En
i=1
AniWni> n/2
4
2
n=1
np22E
ni=1
AniWni En
i=1
AniWni2
(by Chebyshevs inequality)
C
n=1
np22n
i=1
EAniWni2 (by Lemma 2.1 with r = 2)
=C
n=1
np22n
i=1
E|Ani|2EWni
2 (sinceAni is independent ofVni)
C
n=1
np22 n
i=1
E|Ani|2
n2P(V > n) + EV I(V n)2
(by (2.21))
C
n=1
np12EV I(V n)2 +C
n=1
np1P(V> n) (by (4.1))
C
n=1
np12n
i=1
E
V2I
(i 1) < V i
+CEVp (by (4.7))
=C
i=1
n=i
np12E
V2I
(i 1) < V i
+CEVp
C
i=1
ip2E
V2I((i 1) < V i)
+CEVp
C
i=1
ipP(i 1) < V i+CEVp CEVp < .
This proves (4.9). The proof of the theorem is completed.
Remark 4.2.
(i) When > 1, condition (4.2) is automatically satisfied. To see this, for all
n 1,
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Randomly Weighted Sums in Banach Spaces 1773
ni=1
E|Ani|2
n
i=1
(EA2ni)1/22
(by Jensens inequality)
nn
i=1
EA2ni.
So, it follows from (4.1) that
(4.10)
ni=1
E|Ani|= O(n).
Hence
0 limn
nE n
i=1
AniVniI(Vni n)
limn
nn
i=1
EAniVniI(Vni n)
limn
Cn n
i=1
E|Ani|
EV I(V n) +nP(V> n)
(by (2.21))
limn
Cn1EV (by (4.10))
= 0 (since >1 andEV< ).
(ii) WhenX is a real separable Hilbert space andE Vni 0, by similar method(using the identityE(
ni=1 Vni)
2 =n
i=1 EV2
ni instead of Lemma 2.1) we
can prove that Theorem 4.1 still holds without (4.2).
Remark 4.3. When Ani ani a.s., where{ani, n 1, 1 i n} is an arrayof constants satisfying
(4.11) supn1, i1
|ani| < ,
then the condition (4.1) is satisfied automatically. Moreover, the condition (4.2) canbe replaced by
(4.12) nn
i=1
aniVni 0 in probability asn .
To see this, by Lemma 2.2 and (4.12), we can assume that for eachn 1,{Vni, 1 i n} are symmetric random elements. By Remark 4.2, we need only prove (4.2)when0 < 1. We have
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1774 Le Van Thanh and G. Yin
0 limn
nE n
i=1
aniVniI(Vni > n)
limn
nn
i=1
EaniVniI(Vni> n)
limn
Cn n
i=1
|ani|
EV I(V > n) (by (2.22))
limn
Cn1EV I(V> n) (by (4.11))
limn
CEV1/I(V> n) lim
nC
E
VpI(V > n)1/p
(by Jensens inequality and noting that p 1)
= 0 (sinceEVp < ).
It follows that
(4.13) nn
i=1
aniVniI(Vni > n) 0 in probability as n .
Combining (4.12) and (4.13), we have
(4.14) nn
i=1
aniVniI(Vni n) 0 in probability as n .
Now, (4.11) implies that there exists a constant Csuch thatnaniVniI(Vni n) |ani| Ca.s. for all n 1, 1 i n.Hence by Lemma 2.3 and (4.14), (4.2) follows.
Remark 4.4. Ifp 2, then Theorem 4.1 still hold if (4.1) is replaced by
(4.15)
ni=1
E(|Ani|q) =O(n) for some q >2(p 1)/(2 1).
The proof is similar to that of Theorem 4.1, by applying Lemma 2.1 with r = q/2.The verbatim argument is omitted.
Now, if we considerVni aniVni as a linear operator fni : X X, thenwe have the following theorem. The proof is exactly as that of Theorem 4.1 and
Remark 4.4.
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Randomly Weighted Sums in Banach Spaces 1775
Theorem 4.5. Let > 1/2, p 1, {Vni, n 1, 1 i n} be an arrayof row-wise independent random elements that are stochastically dominated by a
random elementV withEVp < . Let{fni, n 1, 1 i n} be an array ofcontinuous linear operators fromX toX. If
(4.16) limn
nE n
i=1
fni(Vni)I(Vni n)= 0,
and if either
(4.17) 1 p < 2,n
i=1
fni2 =O(n),
or
(4.18) p 2,n
i=1
fniq =O(n) for some q >2(p 1)/(2 1),
then
(4.19)
n=1
np2P
ni=1
fni(Vni)
> n
< for all > 0.
By applying the fact that
E max1kn
ki=1
Vni
ni=1
EVni for all arbitrary array {Vni, n 1, 1 i n},
we obtain the following theorem. No row-wise independence conditions are needed.
Theorem 4.6. Let > 0, 0 < p < 1, {Vni, n 1, 1 i n} be an arrayof random elements that are stochastically dominated by a random elementV with
EVp < . Let {Ani, n 1, 1 i n} be an array of random variables.
Assume that for all n 1, 1 i n, the random variable A ni is independent ofVni. If
(4.20)
ni=1
E(|Ani|) =O(n),
then
(4.21)
n=1
np2P
max1kn
ki=1
AniVni> n< for all >0.
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1776 Le Van Thanh and G. Yin
Proof. Let Wni be as in the proof of Theorem 4.1. We need only prove the
theorem forp 1. For all >0,
(4.22)
n=1
np2P
max1kn
k
i=1
AniWni > n
1
n=1
np2E
max1kn
k
i=1
AniWni
(by Chebyshevs inequality)
Cn=1
np2n
i=1
E
AniWni
=Cn=1
np2n
i=1
E|Ani|EWni (sinceAni is independent ofVni)
Cn=1
np2 n
i=1
E|Ani|
nP(V> n)+EV I(V n)
(by (2.21))
Cn=1
np12EV I(V n)2 +Cn=1
np1P(V > n)
(by (4.20))
C
n=1np1
n
i=1 EVI(i 1)
< V i
+ CEVp=C
i=1
n=i
np1E
VI
(i 1) < V i
+C EVp
Ci=1
ipE
VI((i 1) < V i)
+CEVp
Ci=1
ipP
(i 1) < V i
+CEVp
CEVp < .
Similar to (4.7), we also have
(4.23)
n=1 np2Pmax1in Vni > n CEVp < .Since
P
max1kn
ki=1
AniVni> n
P
max1in
Vni> n
+ P
max1kn
k
i=1
AniWni > n
,
the conclusion (4.21) follows from (4.22) and (4.23).
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Randomly Weighted Sums in Banach Spaces 1777
5. COROLLARIES AND EXAMPLES
In this section, we present several corollaries and examples. The first corollary
extends the result of Wang et al. [25] by including larger range for andp.
Corollary 5.1. Let > 1/2, p 1, {Vni, n 1, 1 i n} be an arrayof row-wise independent random elements that are stochastically dominated by a
random elementV withEVp < . Let{ani, n 1, 1 i n} be an array ofconstants satisfying
(5.1) supn1
sup1in
|ani| < .
If
(5.2) n n
i=1
aniVni
0 in probability as n ,then
(5.3)
n=1
np2P n
i=1
aniVni> n< for all > 0.
Proof. SetAni ania.s. Since (4.1) and (4.15) are immediately consequences
of (5.1), Corollary 5.1 follows from Theorem 4.1, Remark 4.3 and Remark 4.4
Remark 5.2. Corollary 5.1 extends the result of Wang et al. [25] and provides
better convergence rates.
(i) We only assume (5.2), whereas Wang et al. [25] assume (1.6).
(ii) We allow p 1, whereas Wang et al. [25] require thatp >1.
Corollary 5.3. Let1/2 t n1/t< for all >0.
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1778 Le Van Thanh and G. Yin
Proof. Letp= 2t, = 2/p= 1/t. Noting that1/2 t 1/2.Therefore Corollary 5.3 follows from Corollary 5.1.
Remark 5.4. Letani n1/tbni. Then Corollary 5.3 implies the p 0,
then
(5.10)
ni=1
E|Ani|r =O(n) for all 0 < r < q.
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Randomly Weighted Sums in Banach Spaces 1779
The following example shows that in Theorem 4.1, we cannot replace (4.1) by the
weaker condition
(5.11)
ni=1
E|Ani|p =O(n).
Example 5.6. Let1 p n=P
|Vnn|>
= 1/ log(n + 1) for all 0 < < 1,
conclusion (4.3) fails.
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Le Van Thanh
Department of Mathematics
Vinh University
Nghe An 42118
Vietnam
E-mail: [email protected]
G. Yin
Department of Mathematics
Wayne State University
Detroit, MI 48202
U.S.A.
E-mail: [email protected]
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