Research report, "Some convergence theorems for arrays of two average index of random elements in Banach spaces with integrable conditions on" pptx


u
m
i=x
m

v
n
j=y
n
A
mnij
V
ij
{V
ij
; i, j ∈ Z}
{A
mnij
; x
m
 i  u
m
, y
n
 j  v
n
, m  1, n  1} {x
m
, m ≥ 1}
{u

m
, m ≥ 1} {y
n
, n ≥ 1} {v
n
, n ≥ 1}
{V
ij
; i, j ∈ Z}
(Ω, F, P) {A
mnij
; x
m
 i  u
m
, y
n
 j  v
n
, m  1, n  1}
{x
n
, n ≥ 1} {u
n
, n ≥ 1} {y
n
, n ≥ 1} {v
n
, n ≥ 1}
u

n
−x
n
> 0 n ≥ 1 u
n
−x
n
→ ∞ n → ∞ v
n
−y
n
> 0
n ≥ 1 v
n
− y
n
→ ∞ n → ∞

u
m

i=x
m
v
n

j=y
n
A
mnij

V
ij

L
p
−→ 0.

v
n
j=u
n
A
nj
V
j
L
1
expected value mean V EV
Pettis integral provided it exists V expected value EV ∈ X if f(EV ) =
Ef (V ) f ∈ X
∗
X
∗
dual
X EV EV  < ∞
1
{A
nj
; u
n

 j  v
n
, n ≥ 1}
{V
j
, j ∈ Z} X
 ·  (Ω, F, P) {F
n
, n ≥ 1} σ
F n ≥ 1 E
F
n
(Y )
Y F
n
{V
j
, j ∈ Z} {A
nj
}
p {F
n
}  > 0 a
o
= a
o
() > 0
sup
n≥1
v

n

j=u
n
|A
nj
|
p
E
F
n
(V
j

p
I(V
j
 > a
o
)) <  a.s.
A
nj
= a
nj
u
n
 j  v
n
, n ≥ 1
F

n
= {∅, Ω} n ≥ 1 {A
nj
} p
{F
n
} {|a
nj
|
p
}
{V
j

p
, j ∈ Z}
Lemma 1. {k
mn
, m ≥ 1, n ≥ 1}
lim
m∨n→∞
k
mn
= ∞ {X
ij
; i, j ∈ Z}
sup
a>0
sup
n≥1

1
k
mn
u
m

i=x
m
v
n

i=y
n
aP{|X
ij
| > a}  M < ∞, (2.1)
and
lim
a→+∞
sup
n≥1
1
k
mn
u
m

i=x
m
v

n

j=y
n
aP{|X
ij
| > a} = 0. (2.2)
1
k
p
mn
u
m

i=x
m
v
n

i=y
n
E(|X
ij
|
p
I(|X
ij
|  k
mn
)) → 0 as m ∨ n → ∞ (p > 1). (2.3)

Proof.
1
k
p
mn
u
m

i=x
m
v
n

j=y
n
E(|X
ij
|
p
I(|X
ij
|  k
mn
)) =
=
1
k
p
mn
u

m

i=x
m
v
n

j=y
n
E(|X
ij
|
p
I(|X
ij
|  1)) +
1
k
p
mn
u
m

i=x
m
v
n

j=y
n

k
mn

l=2
E(|X
ij
|
p
I(l − 1 < |X
ij
|  l ))
=: A
mn
+ B
mn
.
We first verify that
lim
m∨n→∞
A
mn
= 0.
A
mn
=
1
k
p
mn
u

m

i=x
m
v
n

i=y
n
E(|X
ij
|
p
I(|X
ij
|  1))
=
1
k
p
mn
u
m

i=x
m
v
n

j=y

n

∞

l=1
E(|X
ij
|
p
I(
1
l + 1
< |X
ij
| 
1
l
))


1
k
p
mn
u
m

i=x
m
v

n

j=y
n

∞

l=1
1
l
p
P{
1
l + 1
< |X
ij
| 
1
l
}

=
1
k
p
mn
u
m

i=x

m
v
n

j=y
n

∞

l=1
1
l
p

P{|X
ij
| >
1
l + 1
} − P{|X
ij
| >
1
l
}


=
1
k

p
mn
u
m

i=x
m
v
n

j=y
n

∞

l=1

1
l
p
−
1
(l + 1)
p

P{|X
ij
| >
1
l + 1

}

=
1
k
p−1
mn
∞

l=1

1
l
p
−
1
(l + 1)
p

(l + 1)

1
k
mn
u
m

i=x
m
v

n

j=y
n
1
l + 1
P{|X
ij
| >
1
l + 1
}

 M
1
k
p−1
mn
∞

l=1

1
l
p
−
1
(l + 1)
p


(l + 1) (by (2.1))
= M
1
k
p−1
mn
(
∞

l=1
1
l
p
+ 1) → 0 as m ∨ n → ∞. (2.4)
Next, we will show that
lim
m∨n→∞
B
mn
= 0.
In deed, since
k
mn

l=2
(l
p
− (l − 1)
p
)

k
p−1
mn
(l − 1)
=
1
k
p−1
mn
k
mn

l=2
l
p
(l − 1)l
+
k
mn
k
mn
− 1
−
2
p−1
k
p−1
mn

2

k
p−1
mn
k
mn

l=2
l
p−2
+
k
mn
k
mn
− 1
 4,
By (2.2) we have
B
mn
=
1
k
p
mn
u
m

i=x
m
v

n

j=y
n

k
mn

l=2
E(|X
ij
|
p
I(l − 1 < |X
ij
|  l ))


1
k
p
mn
u
m

i=x
m
v
n


j=y
n

k
mn

l=2
l
p
P{l − 1 < |X
ij
|  l }

=
1
k
p
mn
u
m

i=x
m
v
n

j=y
n

k

mn

l=2
l
p
[P{|X
ij
| > l − 1} − P{|X
ij
| > l}]

=
1
k
p
mn
u
m

i=x
m
v
n

j=y
n

k
mn


l=2
[l
p
− (l − 1)
p
]P{|X
ij
| > l − 1}

=
k
mn

l=2

(l
p
− (l − 1)
p
)
k
p−1
mn
(l − 1)

1
k
mn
u
m


i=x
m
v
n

i=y
n
(l − 1)P{|X
ij
| > l − 1}

 4.
1
k
mn
u
m

i=x
m
v
n

j=y
n
(l − 1)P{|X
ij
| > l − 1}
 4. sup

m≥1,n≥1
1
k
mn
u
m

i=x
m
v
n

j=y
n
(l − 1)P{|X
ij
| > l − 1}
→ 0 as l → ∞. (2.5)
So the conclusion (2.3) follows from (2.4) and (2.5).
Corollary 1. {a
mnij
; x
m
 i  u
m
, y
n
 j  v
n
, m ≥ 1, n ≥ 1}

u
m

i=x
m
v
n

j=y
n
|a
mnij
|  M < ∞
and
sup
x
m
iu
m
,y
n
jv
n
|a
mnij
| → 0 as m ∨ n → ∞.
{X
ij
; i, j ∈ Z} {|a
mnij

|}
lim
a→+∞
sup
m≥1,n≥1
u
m

j=x
m
u
n

j=y
n
|a
mnij
|E(|X
ij
|I(|X
ij
| > a)) = 0.
c
mn
=
1
sup
x
m
iu

m
,y
n
jv
n
|a
mnij
|
u
m

i=x
m
v
n

j=y
n
|a
mnij
|
q
E(|X
ij
|
q
I(|X
ij
|  c
mn

)) → 0 as m ∨ n → ∞ (q > 1).
Proof. Applying Lemma 1 with k
mn
= [c
mn
] + 1 and X
ij
is replaced by a
mnij
c
mn
X
ij
.
{Y
n
, n ≥ 1} Bernoulli sequence {Y
n
, n ≥ 1}
P{Y
1
= 1} = P{Y
1
=
−1} = 1/2 X
∞
= X × X × X × . . .
C(X ) = {(v
1
, v

2
, . . .) ∈ X
∞
:
∞

n=1
v
n
v
n
converges in probability}.
1  p  2 X Rademacher
type p C (0 < C < ∞)
E
∞

n=1
Y
n
v
n

p
 C
∞

n=1
v
n


p
for all (v
1
, v
2
, v
3
, . . .) ∈ C(X ). (2.6)
φ 1  p  2
Rademacher type p C (0 < C < ∞)
E




n

j=1
V
j





p
 C
n


j=1
V
j

p
(2.7)
{V
1
, V
2
, . . . , V
n
}
1 < p  2
1  r < p
L
p
l
p
2∧p p ≥ 1
a, b ∈ R, min{a, b} max{a, b} a ∧ b
a∨b C (0 < C < ∞)
Theorem 1. 1  r < p  2 {V
ij
; i, j ∈ Z}
(Ω, F, P)
p X {A
mnij
; x
m

 i  u
m
, y
n
 j  v
n
, m  1, n  1}
u
m

i=x
m
v
n

j=y
n
E|A
mnij
|
r
 M < ∞ (3.1)
and
sup
x
m
iu
m
,y
n

jv
n
E|A
mnij
|
r
→ 0 as m ∨ n → ∞. (3.2)
{F
mn
; m ≥ 1, n ≥ 1} σ F A
mnij
, x
m

i  u
m
, y
n
 j  v
n
F
mn
{V
ij
; i, j ∈ Z} {A
mnij
}
r {F
mn
}

lim
a→+∞
sup
m≥1,n≥1
u
m

i=x
m
v
n

j=y
n
|A
mnij
|
r
E
F
mn
(V
ij

r
I(V
ij
 > a)) = 0 a.s. (3.3)
m ≥ 1 n ≥ 1 {A
mnij

V
ij
; x
m
 i  u
m
, y
n
 j  v
n
}
A
mnij
V
ij
m ≥ 1, n ≥ 1, x
m
 i  u
m
, y
n
 j  v
n





u
m


i=x
m
v
n

j=y
n
A
mnij
V
ij





L
r
−→ 0 as m ∨ n → ∞. (3.4)
Proof. Since (3.3) there exists a
o
> 0 such that
E

u
m

j=x
m

v
n

j=y
n
|A
mnij
|
r
E
F
mn
(V
ij

r
I(V
ij
 > a
o
))

< 1, m ≥ 1, n ≥ 1.
Thus
EA
mnij
V
ij
I(V
ij

 > a
o
) < 1 for all x
m
 i  u
m
, y
n
 j  v
n
, m ≥ 1, n ≥ 1.
(3.5)
For all m ≥ 1, n ≥ 1, x
m
 i  u
m
, y
n
 j  v
n
, (by (3.1) and (3.5) we have
EA
mnij
V
ij
 = EA
mnij
V
ij
I(V

ij
  a
o
) + EA
mnij
V
ij
I(V
ij
 > a
o
)
 a
o
E|A
mnij
| + EA
mnij
V
ij
I(V
ij
 > a
o
) < ∞
implying that E(A
mnij
V
ij
) exists. Set c

mn
=
1
sup
x
m
iu
m
,y
n
jv
n
E|A
mnij
|
r
,
V

mnij
= V
ij
I(V
ij
  c
mn
), V

mnij
= V

ij
I(V
ij
 > c
mn
),
b

mnij
= EV

mnij
, b

mnij
= EV

mnij
.
Observe that for each i and j, x
m
 i  u
m
, y
n
 j  v
n
, then V
ij
= (V


mnij
−b

mnij
)+
(V

mnij
− b

mnij
). And since A
mnij
and V
ij
are independent for each m, n, i, j we have
E(A
mnij
(V

ij
− b

mnij
)) = E(A
mnij
(V

mnij

− b

mnij
)) = 0.
Hence
E





u
m

i=x
m
v
n

j=y
n
A
mnij
V
ij






r
= E





u
m

i=x
m
v
n

j=y
n
A
mnij
(V

mnij
− b

mnij
) +
u
m

i=x

m
v
n

j=y
n
A
mnij
(V

mnij
− b

mnij
)





r
 CE





u
m


i=x
m
v
n

j=y
n
A
mnij
(V

mnij
− b

mnij
)





r
+ CE





u
m


i=x
m
v
n

j=y
n
A
mnij
(V

mnij
− b

mnij
)





r
(by c
r
-inequality)
 C

E






u
m

i=x
m
v
n

j=y
n
A
mnij
(V

mnij
− b

mnij
)





p


r/p
+ CE





u
m

i=x
m
v
n

j=y
n
A
mnij
(V

mnij
− b

mnij
)






r
 C

u
m

i=x
m
v
n

j=y
n
EA
mnij
(V

mnij
− b

mnij
)
p

r/p
+ C
u
m


i=x
m
v
n

j=y
n
EA
mnij
(V

mnij
− b

mnij
)
r
 C

u
m

i=x
m
v
n

j=y
n
E|A

mnij
|
p
E(V
ij

p
I(V
ij
  c
mn
))

r/p
+ C
u
m

i=x
m
v
n

j=y
n
E(A
mnij
V
ij


r
I(V
ij
 > c
mn
)).
Now, by (3.3), for arbitrary  > 0 there exists a
o
> 0 such that for all a ≥ a
o
. We
have
E

sup
m≥1,n≥1
u
m

j=x
m
v
n

j=y
n
|A
mnij
|
r

E
F
mn
(V
ij

r
I(V
ij
 > a))

< . (3.6)
This implies
sup
m≥1,n≥1
u
m

i=x
m
v
n

j=y
n
E|A
mnij
|
r
E(V

ij

r
I(V
ij
 > a)) <  ∀a ≥ a
o
. (3.7)
Note that (3.6) means {V
ij

r
; i, j ∈ Z} is {E|A
mnij
|
r
}-uniformly integrable, and then
by Corollary 1 with q = p/r, X
ij
= V
ij

r
and a
mnij
= E|A
mnij
|
r
we get

u
m

i=x
m
v
n

j=y
n
|A
mnij
|
p
E(V
ij

p
I(V
ij
  c
mn
)) → 0 as m ∨ n → ∞.
On the other hand (3.6) also implies
u
m

i=x
m
v

n

j=y
n
E(A
mnij
V
ij

r
I(V
ij
 > c
mn
) → 0 as m ∨ n → ∞.
Thus
E





u
m

i=x
m
v
n


j=y
n
A
mnij
V
ij





r
→ 0 as m ∨ n → ∞.
The proof is completed.
Theorem 2. 0 < r < 1 {V
ij
; i, j ∈ Z}
{A
mnij
; x
m
 i  u
m
, y
n
 j  v
n
, m  1, n  1}
u
m


i=x
m
v
n

j=y
n
(E|A
mnij
|)
r
 M < ∞ (3.8)
and
sup
x
m
iu
m
,y
n
jv
n
E|A
mnij
| → 0 as m ∨ n → ∞. (3.9)
{F
mn
; m ≥ 1, n ≥ 1} σ F
A

mnij
, x
m
 i  u
m
, y
n
 j  v
n
F
mn
{V
ij
; i, j ∈ Z}
{A
mnij
} r {F
mn
}
(3.3)





u
m

i=x
m

v
n

j=y
n
A
mnij
V
ij





L
r
−→ 0 as m ∨ n → ∞. (3.10)
Proof. By (3.3), for arbitrary  > 0 there exists a > 0 such that
E

sup
m≥1,n≥1
u
m

j=x
m
v
n


j=y
n
|A
mnij
|
r
E
F
mn
(V
ij

r
I(V
ij
 > a))

<

2
,
this implies
u
m

i=x
m
v
n


j=y
n
E(A
mnij
V
ij

r
I(V
ij
 > a)) <

2
, m ≥ 1, n ≥ 1.
On the other hand, since

E





u
m

j=x
m
v
n


j=y
n
A
mnij
V
ij
I(V
ij
  a)





r

1/r
 E





u
m

j=x
m
v
n


j=y
n
A
mnij
V
ij
I(V
ij
  a)






u
m

j=x
m
v
n

j=y
n
E(A
mnij
V
ij

I(V
ij
  a))  a
u
m

j=x
m
v
n

j=y
n
E|A
mnij
|
 a

u
m

j=x
m
v
n

j=y
n
(E|A
mnij

|)
r

sup
x
m
iu
m
,y
n
jv
n
(E|A
mnij
|)
1−r
 aM sup
x
m
iu
m
,y
n
jv
n
(E|A
mnij
|)
1−r
→ 0 as m ∨ n → ∞,

there exists m
o
, n
o
such that for all (m ∨ n ) ≥ (m
o
∨ n
o
),
E





u
m

i=x
m
v
n

j=y
n
A
mnij
V
ij
I(V

ij
  a)





r


2
. (3.11)
Hence,
E





u
m

i=x
m
v
n

j=y
n
A

mnij
V
ij





r
= E





u
m

i=x
m
v
n

j=y
n
A
mnij
V
ij
I(V

ij
  a) +
u
m

i=x
m
v
n

j=y
n
A
mnij
V
ij
I(V
ij
 > a)





r
 E






u
m

i=x
m
v
n

j=y
n
A
mnij
V
ij
I(V
ij
  a)





r
+ E






u
m

i=x
m
v
n

j=y
n
A
mnij
V
ij
I(V
ij
 > a)





r
 E





u

m

i=x
m
v
n

j=y
n
A
mnij
V
ij
I(V
ij
  a)





r
+
u
m

i=x
m
v
n


j=y
n
E(A
mnij
V
ij

r
I(V
ij
 > a)) < 
for all (m ∨ n) ≥ (m
o
∨ n
o
),
which completes the proof.
Remark.
{A
mnij
; x
m
 i  u
m
, y
n
 j  v
n
, m  1, n  1} F

mn
m ≥ 1 n ≥ 1
F
mn
= σ(A
mnij
, x
m
 i  u
m
, y
n
 j  v
n
) F
mn
σ
{A
mnij
; x
m
 i  u
m
, y
n
 j  v
n
, m  1, n  1} m ≥ 1
n ≥ 1
p

37
φ
p 21
p 52

u
m
i=x
m

v
n
j=y
n
A
mnij
V
ij
{V
ij
; i, j ∈ Z}
{A
mnij
; x
m

i  u
m
, y
n

 j  v
n
} {x
m
, m ≥ 1} {u
m
, m ≥ 1} {y
n
, n ≥ 1} {v
n
, n ≥ 1}
13
th