Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2007, Article ID 86909, 15 pages
doi:10.1155/2007/86909
Research Article
Best Possible Sufficient Conditions for Strong Law of Large
Numbers for Multi-Indexed Orthogonal Random Elements
Kuo-Liang Su
Received 26 April 2006; Revised 18 December 2006; Accepted 5 February 2007
Recommended by Andrew Rosalsky
It will be shown and
induced that the d-dimensional indices in the Banach spaces
version conditions n (EXn p / |nα | p ) < ∞ are sufficient to yield lim min 1≤ j ≤d (n j )→∞ (1/
d
|nα |) k≤n j =1 (1 − (k j − 1)/n j )Xk = 0 a.s. for arrays of James-type orthogonal random
elements. Particularly, it will be shown also that there are the best possible sufficient conditions for multi-indexed independent real-valued random variables.
Copyright © 2007 Kuo-Liang Su. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
The laws of large numbers for orthogonal random variables or Banach space-valued
random elements are investigated by several authors. A consequence of RademacherMenshov theorem [2, 3] showed that the sufficient condition for a strong law of large
numbers of a sequence oforthogonal real-valued random variables with 0 means and
(σk2 /k2 ) · [log2 (k + 1)]2 < ∞. Warren and Howell [4] profinite second moments is ∞
k=1
1+α /k 1+α ) · log1+α k < ∞, 0 < ∞ ≤ 1, for strong
posed the sufficient condition ∞
k=1 (EXk convergence of the one-dimensional
B-valued James-type orthogonal random variables.
∞
2 2 2
2
2
Móricz [5] showed that ∞
i=1
k=1 (σik /i k ) · [log2 (i + 1)] [log2 (k + 1)] < ∞ is the necessary condition for the strong convergence for arrays of quasi-orthogonal real-valued
random variables. Móricz [6] obtained a sufficient condition for strong limit theorems
for arrays of quasi-orthogonal real-valued random variables. Móricz
et al. [7] showed that
n
the sufficient condition for the strong convergence of (1/mα nβ ) m
i=1
k=1 Xik for arrays
of orthogonal, type p Banach space-valued random elements is
p ∞ ∞ p
p
EXik log2 (k + 1) < ∞.
· log2 (i + 1)
αp βp
i =1 k =1
i k
(1.1)
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International Journal of Mathematics and Mathematical Sciences
In this paper, the strong laws of large numbers will be investigated for James type of orthogonality in a Banach space. In order to induce d-dimensional case, d > 2, we wish
to investigate
the strongly convergent behavior of a more general Cesaro-type means,
m n
α
β
(1/m n ) i=1 k=1 (1 − (i − 1)/m)(1 − (k − 1)/n)Xik , as m,n → ∞, for arrays of twodimensionally indexed orthogonal random elements in a Banach space of type p, 1 ≤
p ≤ 2, and 1/2 < α, β ≤ 1, though Su [1] showed a case of α = 1 = β. In particular, it will
be proven that the sufficient conditions are also the best possible even for independent
real-valued random variables. The definition for an array of orthogonal random elements
and the formulation of previous results and auxiliary lemmas for orthogonality are given
in Section 2. The major results and their proofs are in Sections 3 and 4, respectively.
2. Preliminaries and auxiliary lemmas
The basic definitions and properties of Banach space-valued random variables (or random elements) are well established in the literature (e.g., [8]). In these preliminaries, we
only introduce the concepts which are necessary and not easy to read in the literature.
Our sense of orthogonality throughout this manuscript is that of James type orthogonality. For elements x and y in a Banach space B, x is said to be James orthogonal to y
(denoted by x ⊥J y) if x ≤ x + t y for all t ∈ ᏾. If B is a Hilbert space, then James
type orthogonality agrees with the usual notion of orthogonality where the inner product is 0 since x + t y 2 = (x + t y,x + t y) = x2 + t 2 y 2 + 2t(x, y) ≥ x2 for all t ∈ ᏾
if and only if (x, y) = 0. However, in a Banach space where the norm is not generated
by an inner product, it is possible for x ⊥J y but y ⊥J x with (x, y) = 0. For instance,
let ᏾2 = {(x1 ,x2 ) : (x1 ,x2 ) = |x1 | + |x2 |, x1 ,x2 ∈ ᏾} and x = (2,0) and y = (2, −2).
Then, it is clear that the usual inner product (x, y) = 4 = 0. Next, x ⊥J y but y ⊥J x
since x + t y = |2 + 2t | + | − 2t | ≥ 2 = x and y + tx = |2 + 2t | + | − 2| = 3 < y = 4
while picking t = −1/2. Therefore, it is not possible to create a notation of orthogonality
with a good geometrical meaning in an arbitrary Banach space without the inner product.
As a result, James-type orthogonality is adopted to circumvent this shortcoming [7].
Let {Xik , i,k ≥ 1} be a double sequence of random elements in the Banach space L p (B)
with zero means, that is, E(Xik ) = 0 for all i, k and finite pth moments, EXik p < ∞ for
all i, k, where · is the norm of the separable Banach space B. The following is the
extended definition for arrays of orthogonal random variables in Banach spaces.
Definition 2.1. An array of random elements {Xik } is orthogonal in L p (B), 1 ≤ p < ∞, if
p
(i) EXik < ∞ ∀i,k,
(ii)
n1 n2
p
n1 +m1 n2 +m2
p
≤ E
,
a
X
a
X
E
π
(i),π
(k)
π
(i),π
(k)
π
(i),π
(k)
π
(i),π
(k)
1
2
1
2
1
2
1
2
i=1 k=1
i=1
(2.1)
k =1
for all arrays {aik } ⊆ ᏾, for all n1 , n2 , m1 , and m2 , and for all permutations π1 , π2 of the
positive integers {1,2,...,m1 + n1 } and {1,2,...,m2 + n2 }, respectively.
In retrospect, a separable Banach space B is of type p (1 ≤ p ≤ 2) ifand only if there
is a constant C > 0 (depending on B only) such that E ni=1 Xi p ≤ C ni=1 EXi p when
{Xi } are independent random elements with zero means and finite pth moments [8]. In
Kuo-Liang Su 3
order to obtain the desired results, a useful version of moment inequality for arrays that
extend the results of Howell and Taylor [9] is listed below.
Proposition 2.2. The following conditions are equivalent:
(i) B is a Banach space of type p, 1 ≤ p ≤ 2,
(ii) for each array {Xmn } of orthogonal random elements in L p (B), there exists a constant C (depending on {Xmn } and B) such that, for all m and n,
m n
p
m n
p
X
≤
C
EXik .
E
ik i=1 k=1
(2.2)
i=1 k=1
The next lemma is from Móricz et al. [7].
Lemma 2.3. If {Xik ; i ≥ 1, k ≥ 1} is an array of orthogonal (in L p (B)) random elements in
a Banach space B of type p for some 1 ≤ p ≤ 2, then
E
a+ j
max 1≤ j ≤m b+n
i=a+1 k=b+1
p p a+m
Xik ≤ C1 log2 2m
b+n
p
EXik (2.3)
i=a+1 k=b+1
for some C1 > 0 and
E
a+ j
max max 1≤ j ≤m 1≤t ≤n b+t
i=a+1 k=b+1
p p
p a+m
≤ C2 log2 2m
Xik log
2n
2
b+n
p
EXik i=a+1 k=b+1
(2.4)
for some C2 > 0.
The following lemma can be derived directly from Proposition 2.2 and Lemma 2.3.
Hence, the proofs are omitted.
Lemma 2.4. Let {Xik } be an array of orthogonal (in L p (B)) random elements in a Banach
space B of type p for some 1 ≤ p ≤ 2 and let {aik } be any array of real numbers, then there
exists positive constants C3 and C4 such that, for all m and n, (a,b ≥ 0; m,n ≥ 1)
(i)
(ii)
a+m
E
p
a+m
b+n
p p
aik EXik ,
aik Xik ≤ C3
i=a+1 k=b+1
i=a+1 k=b+1
a+m b+t
p p a+m
a
X
C
log
2n
E max ≤
ik
ik
4
2
1≤t ≤n b+n
i=a+1 k=b+1
b+n
i=a+1 k=b+1
p p
aik EXik .
(2.5)
We also need two more crucial lemmas as follows; they are extended from Kronecker’s
lemma and Shiryayev [10], respectively, and will be proven in Section 4.
Lemma 2.5 (two-dimensionally indexed version of Kronecker’s lemma). Let {am } and
{bn } be sequences of positive increasing numbers, both am ↑ ∞ and bn ↑ ∞ when m → ∞ and
n → ∞, respectively. Let {xi j ; i, j ≥ 1} be an array of positive numbers such that
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International Journal of Mathematics and Mathematical Sciences
∞ ∞
i=1
j =1 (xi j /ai b j )
converges. Then,
1 xi j
−→ 0 as m −→ ∞,
am i=1 j =1 b j
m
(i)
n
1 xi j
−→ 0
bn i=1 j =1 ai
m
n
(2.6)
1 xi j −→ 0 as m −→ ∞ or n −→ ∞.
am bn i=1 j =1
m
(ii)
as n −→ ∞,
n
Lemma 2.6. A sufficient and necessary condition that ζmn → 0 with probability one as m,n →
∞ is that for any ε > 0,
P sup sup ζik > ε −→ 0 as m −→ ∞, n −→ ∞.
(2.7)
i≥m k≥n
3. Major results
Theorems 3.1 and 3.2 are two-dimensionally indexed versions of strong convergence for
Cesaro-type means for arrays of Banach space-valued random elements and hence their
proofs are more complicated than that in real cases because of the structure of spaces.
random elements with zero
Theorem 3.1. Let {Xik } be an array of orthogonal (in L p (B))
∞
p αp βp
means in a Banach space B of type p, for some 1 ≤ p ≤ 2. If ∞
i =1
k=1 (EXik /i k ) <
∞, 1/2 < α, β ≤ 1. Then,
m n 1 i−1
k−1
=0
1
−
1
−
X
lim ik
m,n→∞ mα nβ
m
n
a.s.
(3.1)
i =1 k =1
Theorem 3.2. Let {Xik } be an array of orthogonal (in L p (B))
random elements with zero
∞
p αp βp
means in a Banach space B of type p, for some 1 ≤ p ≤ 2. If ∞
i =1
k=1 (EXik /i k ) ·
p
[log2 (k + 1)] < ∞, 1/2 < α, β ≤ 1. Then,
m n 1 i−1
1−
Xik lim =0
m,n→∞ mα nβ
m
a.s.
(3.2)
i =1 k =1
The generalization to d-dimensional arrays random elements of the previous two theorems can be obtained easily by the same methods [1]. Theorems 3.3 and 3.4 are to show
that the sufficient conditions in the previous theorems are the best possible conditions
for independent real-valued random variables, since the real line is of type p, 1 ≤ p ≤ 2.
p
Theorem 3.3. If {τik } is an array of nonnegative real numbers such that τik /iα(p−1) kβ(p−1) ≤
∞
p αp βp
1 holds for indices i and k and the condition ∞
i =1
k=1 (τik /i k ) = ∞, for some 1 ≤ p ≤ 2
and 1/2 < α, β ≤ 1, then there exists an array {Xik } of independent real-valued random
Kuo-Liang Su 5
p
variables such that E(Xik ) = 0, E|Xik | p = τik , and
m n 1 i−1
k−1
1−
1−
Xik limsup = ∞.
β
α
m
n
m+n→∞ m n
(3.3)
i =1 k =1
p −1
p
Theorem 3.4. If {τik } is an array of nonnegative real numbers such that τik · log2 (k +
∞
p αp βp
1)/iα(p−1) kβ(p−1) ≤ 1 holds for indices i and k and the condition ∞
i=1
k=1 (τik /i k ) ·
[log2 (k + 1)] p = ∞, for some 1 ≤ p ≤ 2 and 1/2 < α, β ≤ 1, then there exists an array {Xik }
p
of independent real-valued random variables such that E(Xik ) = 0, E|Xik | p = τik , and
n m 1 i−1
= ∞.
1
−
X
limsup ik
α β
m
m+n→∞ m n
(3.4)
i =1 k =1
4. Proofs
Here we will verify Lemmas 2.5 and 2.6 first, then prove the case of d = 2, that is, Theorems 3.1, 3.2, 3.3, and 3.4. We may apply the analogous approaches for the d-dimensional
cases, d > 2.
∞
Proof of Lemma 2.5. (i) First, we have ∞
i=1 (xi j /ai b j ) < ∞ for each j ≥ 1 and
j =1 (xi j /
1. Then by the one-dimensional version Kronecker’s
lemma,
we can
ai b j ) < ∞ for each i ≥
n
conclude that (1/am ) m
i=1 (xi j /b j ) → 0 as m → ∞, for every j, and (1/bn ) j =1 (xi j /ai ) → 0
as n → ∞, for every i. Hence, for any ε1 > 0, choose n > N such that (1/bn ) nj=1 (xi j /ai ) ≤
ε1 /2i for all i. Then, we have
m n
n
m
m
1 x i j 1 x i j ε1
=
≤
< ε1 .
bn i=1 j =1 ai i=1 bn j =1 ai i=1 2i
Similarly, for any ε2 > 0, choose m > M such that (1/am )
then
m
i=1 (xi j /b j )
1 x i j 1 x i j ε2
=
≤
< ε2 .
am i=1 j =1 b j j =1 am i=1 b j j =1 2i
m
n
n
m
(4.1)
≤ ε2 /2 j for all j,
n
(4.2)
(ii) Apparently, for any ε > 0, when m > M (say), we can conclude that
1 xi j
1 xi j ≤
< ε.
am bn i=1 j =1
a m i =1 j =1 b j
m
n
m
n
(4.3)
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International Journal of Mathematics and Mathematical Sciences
Proof of Lemma 2.6. Fix any m ≥ 1, for any ε > 0, let Aεmn = {ω : supi≥m ζin > ε} and
ε
ε = lim Aε = ∞
A
{ω : supi≥m ζin 0, as n → ∞} = ε≥0 Aεm =
n=1 k≥n Amk . Then,
m∞ 1/t n mn
ε
ε
t =1 Am . However, P(Am ) = limn P( k≥n Amk ). Hence, for any fixed m ≥ 1,
⇐⇒ P
i ≥m
∞
⇐⇒ P
0 as n −→ ∞ = P
/
A1/t
=0
m
k ≥n
t ≥ 1, any fixed m ≥ 1
for any ε > 0, any fixed m ≥ 1
Aεmk −→ 0
as n −→ ∞, any fixed m ≥ 1
k ≥n i ≥m
∞
(4.4)
as n −→ ∞, any fixed m ≥ 1
sup sup ζik > ε
as n −→ ∞
−→ 0
k ≥n i ≥m
m=1
Aεm
any fixed m ≥ 1
⇐⇒ P sup sup ζik > ε −→ 0
⇐⇒ P
ε ≥0
t =1
⇐⇒ P A1/t
m = 0,
⇐⇒ P Aεm = 0
0 = P ω : sup ζin ⇐⇒ P sup sup ζik > ε −→ 0
k ≥n i ≥m
as m −→ ∞, n −→ ∞.
Proof of Theorem 3.1. We need some useful arguments in the proof in [1, 7], and Lemmas
2.3 and 2.4. For positive integers u and v, for any ε,
P
sup
m>2u ,n>2v
∞ ∞ ξmn > ε ≤
P max
2r <m≤2r+1 2 <n≤2
r =u s =v
maxs+1 ξmn > ε ,
s
(4.5)
where
1 i−1
1−
β
α
m n i=1 k=1
m
m
ξmn =
n
1−
k−1
Xik .
n
(4.6)
We have
maxr+1 s maxs+1 ξmn ≤ ξ2r ,2s +
r
2 <m≤2
2 <n≤2
3
( j)
Ars ,
(4.7)
j =1
where
A(1)
rs = r maxr+1 ξm,2s − ξ2r ,2s ,
2 <m≤2
A(2)
rs = s maxs+1 ξ2r ,n − ξ2r ,2s ,
2 <n≤2
A(3)
rs
=
max
max ξmn − ξm,2s − ξ2r ,n + ξ2r ,2s .
2r <m≤2r+1 2s <n≤2s+1
(4.8)
Kuo-Liang Su 7
Therefore,
P
ε
ε
( j)
max
max ξmn > ε ≤ P ξ2r ,2s >
+ P Ars > .
4
4
2r ≤m≤2r+1 2s ≤n≤2s+1
j =1
3
(4.9)
First, by Markov’s inequality and Lemmas 2.3 and 2.4,
∞ ∞ ε
P ξ2r ,2s >
r =u s =v
4
p
4
≤
ε
p
4
≤
ε
=
p
4
ε
∞ ∞
2r
2s
p
p 1
1 − i − 1 1 − k − 1 EXik p
Γ1
βsp
αr
p
2 i =1 k =1
2r
2s r =u s =v 2
∞ ∞
u
v
r
v
u
s
r
s
2
2
2
2
2 2
2
2
p
1
Γ1
+
+
+
EXik βsp
αr
p
2
r =u s =v 2
i=1 k=1 i=2u +1 k=1 i=1 k=2v +1 i=2u +1 k=2v +1
Γ1
4
( j)
Buv ,
say, for some Γ1 > 0.
j =1
(4.10)
Using some basic calculation, it follows that
2u
2v
p
2(α+β)p
1
(1)
βp
· αpu βpv
= αp
EXik .
Buv
2 − 1 2 − 1 2 2 i=1 k=1
(4.11)
Next,
(2)
Buv
2(α+β)p
2αp − 1 2βp − 1
p
∞
2v
1 EXik ≤
i=2u +1 k=1
2βpv
iαp
+
p ∞
EXik ∞
i=2u +1 k=2v +1
iαp kβp
.
(4.12)
Similarly,
p
p
2
∞
∞
∞
EXik 2(α+β)p
1 EXik (3)
βp
≤ αp
+
,
Buv
iαp kβp
2 − 1 2 − 1 i=1 k=2v +1 2αpu kβp
i=2u +1 k=2v +1
(4)
Buv
u
p
∞
∞
2
2
p
EXik 1
Xik ≤
=
E
.
αr p 2βsp
iαp kβp
r =u s =v 2
i=2u +1 k=2v +1
i=2u +1 k=2v +1
∞ ∞
r
s
(4.13)
Secondly, since
2r +m
A(1)
rs = r maxr+1 ξm,2s − ξ2r ,2s = max r 1≤m≤2 2 <m≤2
t =2r +1
ξt,2s − ξt−1,2s ,
(4.14)
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International Journal of Mathematics and Mathematical Sciences
where
s
ξt,2s − ξt−1,2s =
2
t aik (t,s)Xik ,
i =1 k =1
aik (t,s) =
1
k−1
1− s
2βs
2
(i − 1)
1
1
1
1
−
−
−
(t − 1)α+1 t α+1
(t − 1)α t α
(4.15)
,
we have
aik (t,s) ≤
1
t α+1 2βs
.
(4.16)
Hence, for some Γ2 ,Γ∗2 > 0,
∞ ∞ r =u s =v
ε
4
P A(1)
rs >
≤
∞ ∞
r =u s =v
≤
≤
ε
t =2 +1
r =u s =v
p ∞ ∞
4
ε
4
≤
ε
p
4
≈
ε
p
4
≤
ε
≤
2r +m
ε
P max r ξt,2s − ξt−1,2s >
4
1≤m≤2 r
p ∞ ∞
4
p
≤
p
4
ε
p
4
ε
r =u s =v
∗
Γ2
∗
Γ2
E
2r +m
p s
s
max
ξt,2 − ξt−1,2 1≤m≤2r r
t =2 +1
r+1
Γ2 log2 2 · 2r
∞ ∞
(r + 1) p
r =u s =v
∞ ∞
2sβp
Γ2
2
∞ ∞ rp
p
m 2s
EXik r+1
2
2
∞ ∞ rp
r =u s =v k =1
2βsp
2
∞ 1
s =v k =1
2βsp
s
Γ∗2
m=2r +1
r =u s=v m=2r +1 i=1 k=1
s
Γ∗2
2βsp
p
(4.17)
m(α+1)p
p
m 2s
r p EXik r+1
2
r =u s =v k =1
Eξm,2s − ξm−1,2s m=2r +1 i=1 k=1
s
∗
p 2
r
m(α+1)p 2βsp
r+1
2
2
+
i=1 m=2r +1
r
2
i=1
2
∞ r
r =u i =1
r+1
2
r+1 2
i=2r +1 m=i
1
2r[(α+1)p−1]
rp
2r[(α+1)p−1]
+
r+1
2
i=2r +1
1
i(α+1)p−1
2
∞
r+1
+
p
EXik m(α+1)p
r =u i=2r +1
p
EXik rp
i(α+1)p−1
p
EXik .
Kuo-Liang Su 9
Since the first term and the second term in the bracket of (4.17) can be expressed, respectively, as
2
∞ r
rp
2r[(α+1)p−1]
r =u i=1
2
∞
r+1
≤
∞
i(α+1)p−1
≤ 2(α+1)p
2r[(α+1)p−1]
∞
≤
r2
i=1 u<r;i<2r
rp
r =u i=2r +1
r2
2r[(α+1)p−1]
i=2u +1 u<r;i<2r+1
u
2
1
i =1
i(α+1)p−1
≤ 2(α+1)p
∞
1
i=2u +1
i(α+1)p−1
+
∞
1
i=2u +1
i(α+1)p−1
,
,
(4.18)
hence,
(4.17) ≤ 2
(α+1)p
≤ 2αp
p
4
ε
p
8
ε
∗
Γ2
Γ∗2
∞
s =v
v
s
2
+
k =1
2
k=2v +1
i=1
p
EXik ∞
u
2
+2
i=2u +1
iαp 2βsp
p
p
2 2
2
∞
1 EXik 1 EXik +
2βv p k=1 i=1 iαp
2βv p k=1 i=2u +1 iαp
v
u
v
p
2u ∞
EXik +
k=2v +1 i=1
iαp kβp
+2
(4.19)
p ∞
EXik ∞
k=2v +1 i=2u +1
iαp kβp
.
Similarly, for some Γ3 > 0,
∞ ∞ r =u s =v
P
A(2)
rs
p
ε
8
>
≤ 2βp
4
ε
Γ3
p
p
2 2
2
∞
1 EXik 1 EXik + αup
2αup k=1 i=1 kβp
2 i=1 k=2v +1 kβp
v
+
u
p
2v
∞
EXik i=2u +1 k=1
iαp kβp
u
+2
∞
p ∞
EXik i=2u +1 k=2v +1
iαp kβp
.
(4.20)
Next, since
maxr+1 s maxs+1 ξmn − ξm,2s − ξ2r ,n + ξ2r ,2s r
2 <m≤2
2 <n≤2
2r +m
≤ max r max s 1≤m≤2 1≤n≤2 r
s +n
2
t1 =2 +1 t2 =2s +1
ξt1 ,t2 − ξ(t1 −1),t2 − ξt1 ,(t2 −1) + ξ(t1 −1),(t2 −1) ,
ξt1 ,t2 − ξ(t1 −1),t2 − ξt1 ,(t2 −1) + ξ(t1 −1),(t2 −1) =
t1 t2
(4.21)
bik t1 ,t2 Xik ,
i =1 k =1
where
bik t1 ,t2 =
1
i−1
1
i−1
α 1 −
1−
−
t1α
t1
t1 − 1
t1 − 1
1
k−1
1
k−1
1
−
,
−
· β 1−
β
t2
t2 − 1
t2 − 1
t2
(4.22)
10
International Journal of Mathematics and Mathematical Sciences
so apparently,
bik t1 ,t2 ≤ (i − 1)(k − 1) · 1
1
≤
.
α+1 β+1
t1 − 1 t2 − 1
t1 t2
β+1
t1α+1 t2
(4.23)
Hence, for some Γ4 > 0,
∞ ∞ r =u s =v
≤
P A(3)
rs >
∞ ∞
r =u s =v
≤
≤
p
4
ε
p
4
ε
p
4
≈
ε
2r +m
P max r max s 1≤m≤2 1≤n≤2 r
Γ4
Γ4
Γ4
(α+β)p
s +n
2
t1 =2 +1 t2 =2s +1
≤2
ε
4
∞ ∞
r =u s =v
∞ ∞
log2 2 · 2r
p
∞ ∞
r =u s =v
16
ε
p
p p
r s
s+1
2
p 2
m n
p
p
EXik (α+1)p
m
n(β+1)p
m=2r +1 n=2s +1 i=1 k=1
r+1
2
r+1
2
2
+
i=1 m=2r +1
Γ4
r
EXik (α+1)p
m
n(β+1)p
m=2r +1 n=2s +1 i=1 k=1
r+1
log2 2 · 2s
(r + 1) p (s + 1) p
r =u s =v
ε
ξt1 ,t2 − ξ(t1 −1),t2 − ξt1 ,(t2 −1) + ξ(t1 −1),(t2 −1) >
4
s+1
2
m n
r+1 2s
2
r+1
2
s+1
2
+
k=1 n=2s +1
i=2r +1 m=i
s+1 2
s+1
2
k=2s +1 n=i
p
EXik (α+1)p
m
n(β+1)p
p
p
EXik 1
Xik + 2
E
2αup 2βv p i=1 k=1
2αup i=1 k=2v +1 kβp
2u
+
2
2v
p
∞
2v
EXik 2βv p i=2u +1 k=1
∞
2u
iαp
+4
p ∞
EXik ∞
i=2u +1 k=2v +1
iαp kβp
.
(4.24)
Combining the results in (4.9), (4.10), (4.19), (4.20), and (4.24), we can conclude that
P[limm,n→∞ ξmn = 0] = 1. Since P[supm≥s,n≥t ξmn > ε] → 0 as s,t → ∞ by applying
Lemmas 2.5 and 2.6 on the right-hand side of the previous inequalities, then the proof is
∞ 2r
2r ∞
2r 2r
completed
bya convention
that
i=1
k =1 = 0 =
i=1
k =1 =
i =1
k=1 and
∞ ∞
∞
∞
i=2r
k=2r = i=1
k=1 if r = −1.
Proof of Theorem 3.2. Similar to the previous proof, we start out from assuming that 2r ≤
m ≤ 2r+1 and 2s ≤ n ≤ 2s+1 with nonnegative integers u and v, and letting
1 i−1
1−
Xik .
α
m nβ i=1 k=1
m
m
τmn =
n
(4.25)
Kuo-Liang Su 11
First, we can write
∞ ∞ r =u s =v
P
∞ ∞ 3
ε
( j)
P τ2r ,2s >
+ Drs ,
maxr+1 s maxs+1 τmn > ε ≤
r
2 ≤m≤2
2 ≤n ≤2
4
r =u s =v
(4.26)
j =1
where
(1)
=
Drs
(2)
Drs
(3)
=
Drs
r =u s =v
r =u s =v
P
r =u s =v
ε
,
4
maxr+1 τm,2s − τ2r ,2s >
r
2 <m≤2
∞ ∞ =
∞ ∞ P
∞ ∞ P
ε
r
r
s
max τ2 ,n − τ2 ,2 > ,
(4.27)
4
2s <n≤2s+1
maxr+1 s maxs+1 τmn − τm,2s − τ2r ,n + τ2r ,2s >
r
2 <m≤2
2 <n≤2
ε
.
4
Comparing (4.9) to (4.24), we can easily obtain that for some Γ5 > 0,
p
∞
EXik p
∞ ∞ ∞
ε
4
P τ2r ,2s >
Γ5
≤
ε
4
r =u s =v
(4.28)
iαp kβp
i=2u +1 k=2v +1
and for some Γ6 ,Γ∗6 > 0,
(1)
Drs
p
4
≤
ε
≤2
αp
Γ6
∞ ∞
r =u s =v
p
8
ε
∗
Γ6
log2 2 · 2
r+1
2
r p
p
m 2s
EXik m=2r +1 i=1 k=1
m(α+1)p 2βsp
p
p
∞
2
2 2
1 EXik 1 EXik +
2βv p i=1 k=1 iαp
2βv p i=2u +1 k=1 iαp
u
+
v
p
2u ∞
EXik i=1 k=2v +1
iαp kβp
v
+2
∞
(4.29)
p ∞
EXik i=2u +1 k=2v +1
iαp kβp
.
Next, we need to examine that
2s +n
τ2r ,t − τ2r ,t−1 ,
max τ2r ,n − τ2r ,2s = s maxs+1 2s <n≤2s+1
2 <n≤2 s
t =2 +1
r
τ2r ,t − τ2r ,t−1 =
2 t
(4.30)
cik (r,t)Xik ,
i =1 k =1
where
⎧
1
i−1
1
⎪
⎪
1
−
⎪
β −
⎨ αr
r
2
2
t
cik (r,t) = ⎪ ⎪
i−1
1
1
⎪
⎩
1− r · β,
2αr
2
t
1
,
(t − 1)β
k = 1,2,...,t − 1,
(4.31)
k = t.
12
International Journal of Mathematics and Mathematical Sciences
Hence, we use basic calculus to obtain
cik (r,t) ≤
β
,
2αr t β (t − 1)β
cik (r,t) ≤
k = 1,...,t − 1,
1
,
2αr t β
k = t.
(4.32)
Following the arguments in the proof of Theorem 3.1, we can get that for some Γ7 ,Γ∗7 > 0,
(2)
Drs
≤
∞ ∞
r =u s =v
p
4
≤
ε
p
4
≈
ε
≤ 2βp
2s +n 2r t
ε
P max s cik (r,t)Xik >
4
1≤n≤2 s
t =2 +1 i=1 k=1
Γ7
∞ ∞
log2 2s+1
p
2αr p
r =u s=v
∞ ∞
2
2
[s + 1] p r =u s=v
p
8
ε
Γ∗7
2αr p
s+1
i=1 n=2s +1 k=1
r
Γ7
r
n
2
2 cik (r,t) p EXik p
s+1
n −1 p
EXik i=1 n=2s +1
p
2u 2v
1 EXik kβp
2αup i=1 k=1
+
p
∞
2v
EXik i=2u +1 k=1
+
∞
n2βp
k =1
iαp kβp
+
p
2u
∞
1 EXik ∞
+
p
∞
EXik i=2u +1 k=2v +1
iαp kβp
(4.33)
kβp
2αup i=1 k=2v +1
p
∞
EXik i=2u +1 k=2v +1
p EXin +
nβp
p
iαp kβp
1 + log2 (k + 1)
.
The second term in (4.33) is obtained by the following basic calculation. Next, similar to
the procedure of getting (4.24), we have
maxr+1 s maxs+1 τmn − τm,2s − τ2r ,n + τ2r ,2s r
2 <m≤2
2 <n≤2
2r +m
≤ max r max s 1≤m≤2 1≤n≤2 r
s +n
2
t1 =2 +1 t2 =2s +1
τt1 ,t2 − τt1 −1,t2 − τt1 ,t2 −1 + τt1 −1,t2 −1 ,
(4.34)
where
τt1 ,t2 − τt1 −1,t2 − τt1 ,t2 −1 + τt1 −1,t2 −1 =
t1 t2
dik t1 ,t2 Xik ,
i =1 k =1
⎧ 1
i−1
1
i−1
1
1
⎪
⎪
1
−
1
−
−
−
⎪
α
β ,
⎪ α
β
t1
t1 − 1
t1 − 1
⎨ t1
t2 − 1
t2
dik t1 ,t2 = ⎪ 1
i−1
1
i−1
1
⎪
⎪
⎪
−
1
−
1
−
,
α
⎩ tα
t1
t1 − 1 t2β
t1 − 1
1
k ≤ n − 1,
k = n.
(4.35)
Kuo-Liang Su 13
Similarly
dik t1 ,t2 ≤
β
dik t1 ,t2 ≤
k ≤ n,
2β ,
t1α+1 t2
1
β,
t1α+1 t2
k = n.
(4.36)
Then, by the analogous approaches of the proof of Theorem 3.1, for some H > 0, we have
(3)
Drs
p
4
≤
ε
H · (lower order terms) +
+
∞
∞
i=2u +1 k=2v +1
p
∞
EXik ∞
iαp kβp
i=2u +1 k=2v +1
p
EXik p
· 1 + log2 (k + 1) .
iαp kβp
(4.37)
Consequently, following the analogous arguments in the previous proof, we can conclude
the desired result.
p
p p
∞
αp βp
Proof of Theorem 3.3. By ∞
{ε } of nonini =1
k=1 (τik /i k ) = ∞, there exists an array
ik∞
p p
creasing positive numbers converging to 0 as max{i,k} → ∞ such that ∞
i =1
k=1 (τik εik /
iαp kβp ) = ∞. Next, define an array of independent random variables {Wik } with the following properties [8]:
P Wik =
τ εik
iα k β
,
= αpik βp
εik
i k
p p
P Wik = 0 = 1 −
τik εik
.
iαp kβp
(4.38)
Then, it is easy to have that
p
E|Wik | p = τik ,
p p −1
0 ≤ E(Wik ) =
τik εik
p −1
≤ εik .
iα(p−1) kβ(p−1)
(4.39)
p
Now let Xik = τik · ((Wik − E(Wik ))/δik ) = Xik+ − Xik− , where δik = E|Wik − E(Wik )| p , Xik+ =
τik Wik /δik , and Xik− = τik E(Wik )/δik . Since Wik and E(Wik ) are all positive, and by the
dominated convergence theorem for δik , we have
τik
≥ 1,
δik
lim
τik
i+k→∞ δik
= 1.
(4.40)
14
International Journal of Mathematics and Mathematical Sciences
p −1
Consequently, we can obtain that Xik+ ≥ max{0,Wik }, i,k = 1,2,3,..., and |Xik− | ≤ 2εik ,
when i + k is sufficiently large. Then,
m n 1 i−1
k−1
1−
1−
Xik mα n β
m
n
i=1 k=1
≥
m
−
n
m
n
m i=[m/2]
C,C1 > 0,
C
mα n β
C
≥ α β
m n
≥
i−1
m
1−
Xik+
1−
n
1 i−1
1−
β
α
m n i =1 k =1
m
C
≥ α β
m n
1 i−1
1−
mα nβ i=1 k=1
m
m
≥
1 i−1
1−
m α n β i =1 k =1
m
k − 1 +
Xik n
1−
k − 1 −
Xik n
1−
p −1 k − 1 + 2ε11 m + 1
Xik − α β
n
m n
2
n
1−
k=[n/2]
k−1
n
+
X[m/2],[n/2]
−
n+1
2
Xik+ ≥ 0
C1 mn
,
mα n β
are nondecreasing when i + k is large enough
m n
C1 mn
· · X+
−
4 4 [m/2],[n/2] mα nβ
m n τ[m/2],[n/2]
C mn
· · ·
W[m/2],[n/2] − 1α β
4 4 δ[m/2],[n/2]
m n
·
≥
C
m n [m/2]α [n/2]β C1 mn
·
· ·
− α β
mα n β 4 4
ε[m/2],[n/2]
m n
≈
C·m·n
C1 mn
C2
m·n
−
≈
·
,
64 · ε[m/2],[n/2] mα nβ ε[m/2],[n/2] mα nβ
C2 > 0,
(4.41)
where [·] denotes the greatest integer function. Finally, employing the Borel-Cantelli
lemma and the assumption conditions can yield that
m n 1 i−1
k−1
= ∞,
limsup 1
−
1
−
X
ik
α β
m
n
m+n→∞ m n
a.s.
(4.42)
i =1 k =1
This completes the desired proof.
Proof of Theorem 3.4. Here we define an array {Wik } of independent random variables as
follows [8]:
p p
τ εik
iα k β
p
log2 (k + 1),
P Wik =
= αpik βp
εik log2 (k + 1)
i k
p p
τ εik
p
P Wik = 0 = 1 − αpik βp
log2 (k + 1).
i k
(4.43)
Kuo-Liang Su 15
Hence, we also have that
p
p
EWik = τ ,
ik
0 ≤ E Wik
p p −1
p −1
τ ε log2 (k + 1)
p −1
≤ εik .
= ik ikα(p−1) β(p−1)
i
k
(4.44)
Then, choosing {Xik } and following the similar steps as in the proof of Theorem 3.3, we
can obtain the desired results.
Acknowledgment
The author would like to thank the referee for his careful reading and valuable comments
which make this paper more clear and readable.
References
[1] K.-L. Su, “Strong limit theorems of Cesaro type means for arrays of orthogonal random elements
with multi-dimensional indices in Banach spaces,” Stochastic Analysis and Applications, vol. 22,
no. 2, pp. 237–250, 2004.
[2] P. Révész, The Laws of Large Numbers, vol. 4 of Probability and Mathematical Statistics, Academic
Press, New York, NY, USA, 1968.
[3] W. F. Stout, Almost Sure Convergence, Academic Press, New York, NY, USA, 1974.
[4] P. Warren and J. Howell, “A strong law of large numbers for orthogonal Banach space-valued
random variables,” in Probability in Banach Spaces (Proc. First Internat. Conf., Oberwolfach,
1975), vol. 526 of Lecture Notes in Math., pp. 253–262, Springer, Berlin, Germany, 1976.
[5] F. Móricz, “Strong limit theorems for quasi-orthogonal random fields,” Journal of Multivariate
Analysis, vol. 30, no. 2, pp. 255–278, 1989.
[6] F. Móricz, “Strong limit theorems for quasi-orthogonal random fields. II,” Acta Scientiarum
Mathematicarum, vol. 55, no. 3-4, pp. 309–316, 1991.
[7] F. Móricz, K.-L. Su, and R. L. Taylor, “Strong laws of large numbers for arrays of orthogonal random elements in Banach spaces,” Acta Mathematica Hungarica, vol. 65, no. 1, pp. 1–16, 1994.
[8] R. L. Taylor, Stochastic Convergence of Weighted Sums of Random Elements in Linear Spaces,
vol. 672 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1978.
[9] J. O. Howell and R. L. Taylor, “Marcinkiewicz-Zygmund weak laws of large numbers for unconditional random elements in Banach spaces,” in Probability in Banach Spaces, III (Medford,
Mass., 1980), vol. 860 of Lecture Notes in Math., pp. 219–230, Springer, Berlin, Germany, 1981.
[10] A. N. Shiryayev, Probability, vol. 95 of Graduate Texts in Mathematics, Springer, New York, NY,
USA, 1984, translated by R. P. Boas.
Kuo-Liang Su: Department of Business, National Open University, Lu Chow 247, Taipei, Taiwan
Email address: klsu@mail.nou.edu.tw