# Statistical Summability through de la Vallée-Poussin Mean in Probabilistic Normed Spaces

Authors: Ayhan Esi
Journal: International Journal of Mathematics and Mathematical Sciences
 (2014)

---

## Abstract

Two concepts—one of statistical convergence and the other of de la Vallée-Poussin mean—play an important role in recent research on summability theory. In this work we define a new type of summability methods and statistical completeness involving the ideas of de la Vallée-Poussin mean and statistical convergence in the framework of probabilistic normed spaces.


---

## Body

## 1. Introduction, Definitions, and Preliminaries

Fast [1] presented the following definition of statistical convergence for sequences of real numbers. Let K
⊆
ℕ, the set of natural numbers, and K
n
=
{
k
≤
n
:
k
∈
K
}. The natural density of K is defined by δ
(
K
)
=
lim
⁡
n
n
-
1
|
K
n
| if the limit exists, where |
K
n
| denotes the cardinality of K
n.The sequencex
=
(
x
j
) is said to be statistically convergent to the number ℓ if for every ϵ
>
0 the set K
ɛ
:
=
{
k
∈
ℕ
:
|
x
k
-
ℓ
|
≥
ɛ
} has natural density zero; that is, for each ϵ
>
0,
(1)
lim
⁡
n
1
n
|
{
j
≤
n
:
|
x
j
-
ℓ
|
≥
ϵ
}
|
=
0
.
Note that every convergent sequence is statistically convergent to the same limit, but its converse need not be true.In 1985, Fridy [2] has defined the notion of statistically Cauchy sequence and proved that it is equivalent to statistical convergence and since then a large amount of work has appeared. Various extensions, generalizations, variants, and applications have been given by several authors so far, for example, [3–8] and references therein. In the recent past, Mursaleen [9] presented a generalization of statistical convergence by using de la Vallée-Poussin mean which is known λ-statistical convergence and further studied by Çolak and Bektas [10, 11]. For more details related to this concept we refer to [12–18].Letλ
=
(
λ
n
) be a nondecreasing sequence of positive numbers tending to ∞ such that
(2)
λ
n
+
1
≤
λ
n
+
1
,
λ
1
=
0
.
The generalized de la Vallée-Poussin mean is defined by
(3)
t
n
(
x
)
=
:
1
λ
n
∑
j
∈
I
n
‍
x
j
,
where I
n
=
[
n
-
λ
n
+
1
,
n
].A sequencex
=
(
x
j
) is said to be (
V
,
λ
)-summable to a number ℓ if
(4)
t
n
(
x
)
⟶
ℓ
as
n
⟶
∞
.
In this case ℓ is called λ-limit of x.LetK
⊆
ℕ be a set of positive integers; then
(5)
δ
λ
(
K
)
=
lim
⁡
n
1
λ
n
|
{
n
-
λ
n
+
1
≤
j
≤
n
:
j
∈
K
}
|
is said to be λ-density of K.In caseλ
n
=
n, λ-density reduces to the natural density. Also, since (
λ
n
/
n
)
≤
1, δ
(
K
)
≤
δ
λ
(
K
) for every K
⊆
ℕ.The number sequencex
=
(
x
j
) is said to be λ-statistically convergent to the number ℓ if, for each ϵ
>
0, δ
λ
(
K
ϵ
)
=
0, where K
ϵ
=
{
j
∈
ℕ
:
|
x
j
-
ℓ
|
>
ϵ
}; that is,
(6)
lim
⁡
n
1
λ
n
|
{
j
∈
I
n
:
|
x
j
-
ℓ
|
>
ϵ
}
|
=
0
.
In this case we write S
λ
-
lim
⁡
j
x
j
=
ℓ and we denote the set of all λ
-statistically convergent sequences by S
λ.A distribution function is an element ofΔ
+, where Δ
+
=
{
f
:
ℝ
→
[
0,1
]
;
f is left-continuous, nondecreasing, f
(
0
)
=
0 and f
(
+
∞
)
=
1
} and the subset D
+
⊆
Δ
+ is the set D
+
=
{
f
∈
Δ
+
;
l
-
f
(
+
∞
)
=
1
}. Here l
-
f
(
+
∞
) denotes the left limit of the function f at the point x. The space Δ
+ is partially ordered by the usual pointwise ordering of functions; that is, f
≤
g if and only if f
(
x
)
≤
g
(
x
) for all x
∈
ℝ.A triangle function is a binary operation onΔ
+, namely, a function τ
:
Δ
+
×
Δ
+
→
Δ
+ that is associative, commutative nondecreasing and which has e as unit; that is, for all f
,
g
,
h
∈
Δ
+, we have(i)
τ
(
τ
(
f
,
g
)
,
h
)
=
τ
(
f
,
τ
(
g
,
h
)
),
(ii)
τ
(
f
,
g
)
=
τ
(
g
,
f
),
(iii)
τ
(
f
,
h
)
=
τ
(
g
,
h
) whenever f
≤
g,
(iv)
τ
(
f
,
e
)
=
f.Heree is the d.f. defined by
(7)
e
(
x
)
=
{
0
if
x
≤
0
;
1
if
x
>
0
.
We remark that the set Δ as well as its subsets can be partially ordered by the usual pointwise order: in this order, e is the maximal element in Δ
+.There are two definitions of probabilistic normed space, the original one by Šerstnev [19] who used the idea of Menger [20] to define such space and the other one by Alsina et al. [21].According to Šerstnev, a probabilistic normed space (for short, PN-space) is a triple(
X
,
ν
,
τ
), where X is a real linear space, τ is a triangle function, and ν is the probabilistic norm; that is, ν is a map from X into Δ
+ that satisfies the following conditions:(i)
ν
x
=
e if and only if x
=
θ, where θ is the null vector of X,
(ii)
ν
α
x
(
t
)
=
ν
x
(
t
/
|
α
|
) for all t
>
0, α
∈
ℝ with α
≠
0, and x
∈
X,
(iii)
ν
x
+
y
≥
τ
(
ν
x
,
ν
y
) whenever x
,
y
∈
X.Hereν
x
(
t
) denotes the value of ν
x at t
∈
ℝ.In this paper, using the notions of statistical convergence and de la Vallée-Poussin mean, we define and study a new type of summability methods in the setting of probabilistic normed spaces. We also introduce a new type of statistical completeness through de la Vallée-Poussin mean in this framework.


## 2. Statistical Summability through de la Vallée-Poussin Mean

Here we introduce the notions ofλ-summable and statistically λ-summable in PN-space and give some of its properties. We will assume throughout this paper that (
X
,
ℱ
,
τ
) is a probabilistic normed space.Definition 1.
A sequencex
=
(
x
k
) is said to be λ-summable in PN-space (
X
,
ν
,
τ
) or simply (
λ
)
ν-summable to ξ if for each ɛ
>
0, θ
∈
(
0,1
) there exists a positive integer j
0 such that ν
t
j
(
x
)
-
ξ
(
ɛ
)
>
1
-
θ for all j
≥
j
0. In this case one writes ν
(
λ
)
-lim
x
k
=
ξ and ξ is called the ν
(
λ
)-limit of the sequence x
=
(
x
k
).Definition 2.
A sequencex
=
(
x
k
) is said to be statistically λ-summable in (
X
,
ν
,
τ
) or simply S
(
λ
)
ν-summable to ξ if for each ɛ
>
0, θ
∈
(
0,1
) the set K
ɛ
(
λ
)
=
{
j
∈
ℕ
:
ν
t
j
(
x
)
-
ξ
(
ɛ
)
≤
1
-
θ
} has natural density zero (briefly, δ
(
K
ɛ
(
λ
)
)
=
0
); that is,
(8)
lim
⁡
n
1
n
|
{
j
≤
n
:
ν
t
j
(
x
)
-
ξ
(
ɛ
)
≤
1
-
θ
}
|
=
0
.
In this case one writes ν
(
S
λ
)
-lim
x
k
=
ξ, and ξ is called the ν
(
S
λ
)-limit of x. One may write (8) in the alternative form as
(9)
lim
⁡
n
1
n
|
{
j
≤
n
:
ν
t
j
(
x
)
-
ξ
(
ɛ
)
>
1
-
θ
}
|
=
1
.Theorem 3.
If a sequencex
=
(
x
k
) is statistically λ-summable in PN-space, that is, ν
(
S
λ
)
-lim
x
k
=
ξ exists, then it is unique.Proof.
Suppose that there exist two elementsξ
1
,
ξ
2
∈
X with ξ
1
≠
ξ
2 such that ν
(
S
λ
)
-lim
x
k
=
ξ
1 and ν
(
S
λ
)
-lim
x
k
=
ξ
2. Let ϵ
>
0 be given. Choose q
>
0 such that
(10)
τ
(
(
1
-
q
)
,
(
1
-
q
)
)
>
1
-
ϵ
.
Then, for any t
>
0, we define
(11)
M
q
′
(
λ
)
=
{
j
∈
ℕ
:
ν
t
j
(
x
)
-
ξ
1
(
t
)
≤
1
-
q
}
,
M
q
′′
(
λ
)
=
{
j
∈
ℕ
:
ν
t
j
(
x
)
-
ξ
2
(
t
)
≤
1
-
q
}
.
Since ν
(
S
λ
)
-lim
x
=
ξ
1 implies δ
(
M
q
′
(
λ
)
)
=
0 and, similarly, we have δ
(
M
q
′′
(
λ
)
)
=
0. Now, let M
q
(
λ
)
=
M
q
′
(
λ
)
∩
M
q
′′
(
λ
). It follows that δ
(
M
q
(
λ
)
)
=
0 and hence the complement M
q
c
(
λ
) is nonempty set and δ
(
M
q
c
(
λ
)
)
=
1. Now, if k
∈
ℕ
∖
M
q
(
λ
), then
(12)
ν
ξ
1
-
ξ
2
(
t
)
≥
τ
(
ν
t
j
(
x
)
-
ξ
1
(
t
2
)
,
ν
t
j
(
x
)
-
ξ
2
(
t
2
)
)
>
τ
(
(
1
-
q
)
,
(
1
-
q
)
)
>
1
-
ϵ
.
Since ϵ
>
0 was arbitrary, we get ν
ξ
1
-
ξ
2
(
t
)
=
1 for all t
>
0. Hence ξ
1
=
ξ
2. This means that ν
(
S
λ
)-limit is unique.The following theorem gives the algebraic properties of statisticallyλ-summable sequences in PN-spaces.Theorem 4.
Letx
=
(
x
k
) and y
=
(
y
k
) be two sequences. If ν
(
S
λ
)
-lim
x
k
=
ξ
1 and ν
(
S
λ
)
-lim
y
k
=
ξ
2, then(i)
ν
(
S
λ
)
-lim
(
x
k
±
y
k
)
=
ξ
1
±
ξ
2,
(ii)
ν
(
S
λ
)
-lim
α
x
k
=
α
ξ, α
(
≠
0
)
∈
ℝ.Proof of the theorem is straightforward and so omitted.Theorem 5.
If a sequencex
=
(
x
k
) is λ-summable to ξ in PN-space; then it is statistically λ-summable to the same limit.Proof.
Letν
(
λ
)
-lim
x
k
=
ξ. Then for every ϵ
>
0 and t
>
0, there is a positive integer j
0 such that
(13)
ν
t
j
(
x
)
-
ξ
(
t
)
>
1
-
ϵ
for all j
≥
j
0. Since the set
(14)
K
ϵ
(
λ
)
:
=
{
j
∈
ℕ
:
ν
t
j
(
x
)
-
ξ
(
t
)
≤
1
-
ϵ
}
is contained in {
1,2
,
3
,
…
,
j
0
-
1
}. As we know, every finite subset of ℕ has natural density zero; that is, δ
(
K
ϵ
(
λ
)
)
=
0. Hence, a sequence x
=
(
x
k
) is S
(
λ
)
ν-summable to ξ.Remark 6.
The converse of the above theorem is not true in general, which is verified by the following example.Example 7.
Let(
ℝ
,
|
·
|
) denote the space of all real numbers with the usual norm and τ
(
a
,
b
)
=
a
b for all a
,
b
∈
[
0,1
]. Let ν
x
(
t
)
=
t
/
(
t
+
|
x
|
) for all x
∈
X and t
>
0. In this case, we observe that (
ℝ
,
ν
,
τ
) is a PN-space. Define a sequence x
=
(
x
k
) by
(15)
t
j
(
x
)
=
{
j
;
if
j
=
n
2
,
n
∈
ℕ
0
;
otherwise
.
For ϵ
>
0 and t
>
0, write
(16)
K
ϵ
(
λ
)
=
{
j
∈
ℕ
:
ν
t
j
(
x
)
(
t
)
≤
1
-
ϵ
}
.
It is easy to see that
(17)
ν
t
j
(
x
)
(
t
)
=
t
t
+
|
t
j
(
x
)
|
=
{
t
t
+
j
,
for
j
=
n
2
,
n
∈
ℕ
;
1
,
otherwise
;
hence
(18)
lim
⁡
ν
t
j
(
x
)
(
t
)
=
{
0
,
for
if
j
=
n
2
,
n
∈
ℕ
;
1
,
otherwise
.
Therefore, the sequence (
x
k
) is not (
λ
)
ν-summable. But the set K
ϵ
(
λ
) has natural density zero since K
ϵ
(
λ
)
⊂
{
1,4
,
9,16
,
…
}. From here, we conclude that the converse of Theorem 5 need not be true.Theorem 8.
A sequencex
=
(
x
k
) is statistically λ-summable in PN-space to ξ if and only if there exists a subset K
=
{
k
1
<
k
2
<
⋯
<
k
n
<
⋯
}
⊆
ℕ such that δ
(
K
)
=
1 and ν
(
λ
)
-lim
x
k
n
=
ξ.Proof.
Suppose that there exists a subsetK
=
{
k
1
<
k
2
<
⋯
<
k
n
<
⋯
}
⊆
ℕ such that δ
θ
(
K
)
=
1 and ν
(
λ
)
-lim
x
k
n
=
ξ. Then there exists a positive integer N
∈
ℕ such that for n
≥
N
(19)
ν
t
n
(
x
)
-
ξ
(
t
)
>
1
-
ϵ
.
Put K
ϵ
(
λ
)
=
{
n
∈
ℕ
:
ν
t
n
(
x
)
-
ξ
(
t
)
≤
1
-
ϵ
} and K
′
=
{
k
N
+
1
,
k
N
+
2
,
…
}. Then δ
(
K
′
)
=
1 and K
ϵ
(
λ
)
⊆
ℕ
-
K
′ which implies that δ
(
K
ϵ
(
λ
)
)
=
0. Hence x
=
(
x
k
) is statistically λ-summable to ξ in PN-space.
Conversely, let sequencex
=
(
x
k
) is statistically λ-summable to ξ. For q
=
1,2
,
3
,
… and t
>
0, write
(20)
K
q
(
λ
)
=
{
j
∈
ℕ
:
ν
t
k
j
(
x
)
-
ξ
(
t
)
≥
1
-
1
q
}
,
M
q
(
λ
)
=
{
j
∈
ℕ
:
ν
t
k
j
(
x
)
-
ξ
(
t
)
<
1
-
1
q
}
.
Then δ
(
K
q
(
λ
)
)
=
0 and
(21)
M
1
(
λ
)
⊃
M
2
(
λ
)
⊃
⋯
M
i
(
λ
)
⊃
M
i
+
1
(
λ
)
⊃
⋯
,
(22)
δ
(
M
q
(
λ
)
)
=
1
,
q
=
1,2
,
…
.
Now we have to show that forj
∈
M
q
(
λ
), x
=
(
x
k
j
) is (
λ
)
ν-summable to ξ. Suppose that x
=
(
x
k
j
) is not (
λ
)
ν-summable to ξ. Therefore there is ϵ
>
0 such that ν
t
k
j
-
ξ
(
t
)
≥
ϵ for infinitely many terms. Let
(23)
M
ϵ
(
λ
)
=
{
j
∈
ℕ
:
ν
t
k
j
-
ξ
(
t
)
<
ϵ
}
,
and ϵ
>
1
/
q with q
=
1,2
,
3
…. Then
(24)
δ
(
M
ϵ
(
λ
)
)
=
0
,
and, by (21), M
q
(
λ
)
⊂
M
ϵ
(
λ
). Hence δ
(
M
q
(
λ
)
)
=
0, which contradicts (22) and therefore x
=
(
x
k
j
) is (
λ
)
ν-summable to ξ.Similarly we can prove the following dual statement.Theorem 9.
A sequencex
=
(
x
k
) is λ-statistically summable in PN-space to ξ if and only if there exists a subset K
=
{
k
1
<
k
2
<
⋯
<
k
n
<
⋯
}
⊆
ℕ such that δ
λ
(
K
)
=
1 and ν
-lim
x
k
n
=
ξ.


## 3. Statistically Complete through de la Vallée-Poussin Mean

In this section, we define the notions of statisticallyλ-Cauchy and statistically λ-complete with respect to probabilistic normed space and prove related results.Definition 10.
A sequencex
=
(
x
k
) is said to be statistically λ-Cauchy in (
X
,
ν
,
τ
) or simply S
(
λ
)
ν-Cauchy if, for every ϵ
>
0 and θ
∈
(
0,1
), there exists a number N
=
N
(
ϵ
) such that, for all j
,
h
≥
N, the set S
ϵ
(
λ
)
=
{
j
∈
ℕ
:
ν
t
j
(
x
)
-
t
h
(
x
)
(
ϵ
)
≤
1
-
θ
} has natural density zero (briefly, δ
(
S
ϵ
(
λ
)
)
=
0
); that is,
(25)
lim
⁡
n
1
n
|
{
j
≤
n
:
ν
t
j
(
x
)
-
t
h
(
x
)
(
ϵ
)
≤
1
-
θ
}
|
=
0
.Theorem 11.
A sequencex
=
(
x
k
) is statistically λ-summable in PN-space; then it is statistically λ
-Cauchy.Proof.
Suppose thatν
(
S
λ
)
-lim
x
k
=
ℓ. Let ϵ
>
0 be a given number and choose q
>
0 such that
(26)
τ
(
(
1
-
q
)
,
(
1
-
q
)
)
>
1
-
ϵ
.
Then, for t
>
0, we have δ
(
A
q
(
λ
)
)
=
0, where A
q
(
λ
)
=
{
j
∈
ℕ
:
ν
t
j
(
x
)
-
ℓ
(
t
/
2
)
≤
1
-
q
} which implies that
(27)
δ
(
A
q
c
(
λ
)
)
=
δ
(
{
j
∈
ℕ
:
ν
t
j
(
x
)
-
ℓ
(
t
2
)
>
1
-
q
}
)
=
1
.
Let m
∈
A
q
c
(
λ
). Then ν
t
m
(
x
)
-
ℓ
(
t
/
2
)
>
1
-
q.
Now, let(28)
B
ϵ
(
λ
)
=
{
j
∈
ℕ
:
ν
t
j
(
x
)
-
t
m
(
x
)
(
t
)
≤
1
-
ϵ
}
.
We need to show that B
ϵ
(
λ
)
⊂
A
q
(
λ
). Let j
∈
B
ϵ
(
λ
). Then ν
t
j
(
x
)
-
t
m
(
x
)
(
t
)
≤
1
-
ϵ and hence ν
t
j
(
x
)
-
ℓ
(
t
/
2
)
≤
1
-
q; that is, j
∈
A
q
(
λ
). Otherwise, if ν
t
j
(
x
)
-
ℓ
(
t
/
2
)
>
1
-
q, then
(29)
1
-
ϵ
≥
ν
t
j
(
x
)
-
t
m
(
x
)
(
t
)
≥
τ
(
ν
t
j
(
x
)
-
ℓ
(
t
2
)
,
ν
t
m
(
x
)
-
ℓ
(
t
2
)
)
>
τ
(
(
1
-
q
)
,
(
1
-
q
)
)
>
1
-
ϵ
,
which is not possible. Therefore B
ϵ
(
λ
)
⊂
A
q
(
λ
) and hence a sequence x is statistically λ-Cauchy in PN-space.Definition 12.
Let(
X
,
ν
,
τ
) be a PN-space. Then,(i)
PN-space is said to be complete if every Cauchy sequence is convergent in(
X
,
ν
,
τ
);
(ii)
PN-space is said to be statisticallyλ-complete or simply S
(
λ
)
ν-complete if every statistically λ-Cauchy sequence in (
X
,
ν
,
τ
) is statistically λ-summable.Theorem 13.
Every probabilistic normed space(
X
,
ν
,
τ
) is statistically λ-complete but not complete in general.Proof.
Suppose thatx
=
(
x
k
) is statistically λ-Cauchy in PN-space but not statistically λ-summable. Then there exists M
∈
ℕ such that
(30)
δ
(
E
ϵ
(
λ
)
)
=
δ
(
{
j
∈
ℕ
:
ν
t
j
(
x
)
-
t
m
(
x
)
(
t
)
≤
1
-
ϵ
}
)
=
0
,
δ
(
F
ϵ
(
λ
)
)
=
δ
(
{
j
∈
ℕ
:
ν
t
j
(
x
)
-
ℓ
(
t
2
)
>
1
-
ϵ
}
)
=
0
.
This implies that δ
(
F
ϵ
c
(
λ
)
)
=
1. Since
(31)
ν
t
j
(
x
)
-
t
m
(
x
)
(
t
)
≥
2
ν
t
j
(
x
)
-
ℓ
(
t
2
)
>
1
-
ϵ
,
if ν
t
j
(
x
)
-
ℓ
(
t
/
2
)
>
(
1
-
ϵ
)
/
2, then δ
(
E
ϵ
c
(
λ
)
)
=
0; that is, δ
(
E
ϵ
(
λ
)
)
=
1, which leads to a contradiction, since x
=
(
x
k
) was statistically λ
-Cauchy. Hence x
=
(
x
k
) must be statistically λ
-summable in PN-space.
A probabilistic normed space is not complete in general; we verify this by the following example.
Example
14. Let X
=
(
0,1
) and ν
x
(
t
)
=
t
/
(
t
+
|
x
|
) for t
>
0. Then (
X
,
ν
,
τ
) is a probabilistic normed space but not complete, since the sequence (
1
/
(
n
+
1
)
) is Cauchy with respect to (
X
,
ν
,
τ
) but not convergent with respect to the present PN-space.



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*Source: 674159-2014-04-16.xml*