International Journal of Analysis and Applications

Volume 17, Number 5 (2019), 711-721

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-17-2019-711

GENERALIZED STATISTICAL CONVERGENCE OF DOUBLE SEQUENCES IN

PARANORMED SPACES

ABDULLAH ALOTAIBI1,2, ALAA MOHAMMED ALJAHILI2 AND S. A.

MOHIUDDINE1,∗

1Operator Theory and Applications Research Group, Department of Mathematics, Faculty of Science, King

Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

2Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia

∗Corresponding author: mohiuddine@gmail.com

Abstract. We introduce the notion of (λ,µ)-statistical convergence of double sequences in a setting of

paranormed space and prove that every convergent sequence is (λ,µ)-statistically convergent but not con-

versely by supporting an illustrative example. We also define the notions of (λ,µ)-statistical Cauchy and

strongly (λ,µ)p-summable double sequences in a paranormed space and obtain their relationship with (λ,µ)-

statistical convergence.

1. Introduction and preliminaries

The notion of statistical convergence, which is an extension of the idea of common convergence, was

first appeared, under the name of almost convergence, in the first edition of the celebrated monograph of

Zygmund [32]. This idea was introduced by Fast [11] as follows: The sequence x = (xk) is statistically

convergent to ` if for every ε > 0, limn n
−1|{k ≤ n : |xk − `| ≥ ε}| = 0. Some basic properties of statistical

convergence were studied by Schoenberg [30], S̆alát [31] and Connor [9]. An interesting notion of statistically

Received 2019-01-17; accepted 2019-02-21; published 2019-09-02.

2010 Mathematics Subject Classification. 40A05, 40G05.

Key words and phrases. double sequence; statistical convergence; statistical Cauchy; strong p-Cesàro summability; para-

normed space.

c©2019 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

711

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-17-2019-711


Int. J. Anal. Appl. 17 (5) (2019) 712

Cauchy sequence was first defined by Fridy [12] and he also showed that it is equivalent to statistical

convergence. Thereafter, this notion turned out to be one of the most active areas of research in summability

theory. The statistical convergence was studied in various setup such as topological Hausdorff groups [8],

normed spaces [15], locally convex Hausdorff topological spaces [16], paranormed spaces [2], random 2-normed

spaces [19] and many others. Mursaleen [24] presented a generalization of statistical convergence with the

help of non-decreasing sequence λ = (λk) such that λk+1 ≤ λk + 1 and λ1 = 0 of positive numbers tending to

∞ and called it λ-statistical convergence. We also refer to the recent work in [1, 3, 4, 6, 7, 10, 14, 17, 18, 21, 22]

for some applications of convergence methods to approximation theorems. Pringsheim [29] extended the

notion of usual convergence from single sequences of real numbers to double sequences as follows: A double

sequence x = (xjk) has a Pringsheim limit ξ (convergent to ξ in Pringsheim’s sense), in symbols, we shall

write (P) lim x = ξ, provided that given an � > 0 there exists an N ∈ N such that |xjk − ξ| < � whenever

j,k > N. Also, x = (xjk) is bounded, denoted by L∞, if

‖x‖ = sup
j,k
|xjk| < ∞.

It is well known that every convergent single sequence is bounded but this fact need not be true for double

sequences. Statistical convergence extended to double sequences by Mursaleen and Edely [26] with the help

of two dimensional analogue of natural density of subsets of N × N and further it was defined and studied

in intuitionistic fuzzy normed spaces, locally solid Riesz spaces and paranormed spaces by Mursaleen and

Mohiuddine [27, 28], Mohiuddine et al. [20] and Arani et al. [5], respectively. Also, we refer to [13, 23]. Let

K ⊂ N×N. Then, the double natural density of K is defined by

δ2(K) = (P) lim
m,n

|K(m,n)|
mn

provided that the limit exists, where |K(m,n)| be the numbers of (j,k) in K such that j ≤ m and k ≤ n.

x = (xjk) is said to be statistically convergent to ξ if for each � > 0,

(P) lim
m,n

(mn)−1|{(j,k),j ≤ m,k ≤ n : |xjk − ξ| ≥ �}| = 0.

Mursaleen et al. [25] defined and studied the notion of (λ,µ)-statistical convergence for double sequences

where λ = λm and µ = µn are two non-decreasing sequences of positive real numbers each tending to ∞

such that λ1 = 0, λm+1 ≤ λm + 1 and µ1 = 0, µn+1 ≤ µn + 1 for all m,n. The (λ,µ)-density of the set

K ⊆ N×N is given by

δλ,µ(K) = (P) lim
m,n

1

λmµn
|{m−λm + 1 ≤ j ≤ m,n−µn + 1 ≤ k ≤ n : (j,k) ∈ K}|



Int. J. Anal. Appl. 17 (5) (2019) 713

provided that the limit exists. We remark that λm = m and µn = n, the (λ,µ)-density reduces to the natural

double density. A double sequence x = (xjk) is (λ,µ)-statistically convergent to ξ if for every � > 0,

(P) lim
m,n

1

λmµn
|{(j,k),j ∈ Im,k ∈ Jn : |xjk − ξ| ≥ �}| = 0

where Im = [m−λm + 1,m] and Jn = [n−µn + 1,n].

If X is a linear space and g : X → R such that (i) x = 0 ⇒ g(x) = 0, (ii) g(x + y) ≤ g(x) + g(y), (iii)

g(−x) = g(x) and (iv) if tk → t (k → ∞) and xk → x (k → ∞) in the sense that g(xk −x) → 0 (k → ∞)

for scalars tk, t and the vectors xk,x ∈ X, then tkxk → tx (k → ∞) in the sense that g(tkxk − tx) → 0

(k →∞), then g is said to be a paranorm on X and the pair (X,g) is called a paranormed space. Note that

if g(x) = 0 implies x = 0, then paranorm g is called a total paranorm on X and the pair (X,g) is called a

total paranormed space. It is to further note that each seminorm (norm) on X is a paranorm (total) but

not conversely.

2. (λ,µ)-Statistical convergence in paranormed spaces

In this section, we introduce the notion of convergence and (λ,µ)-statistical convergence in the frame-

work of paranormed space and prove various interesting results and display an illustrative example is support

of our result.

Definition 2.1. Let (X,g) be a paranormed space. We say that a double sequence x = (xjk) is convergent,

shortly, g2-convergent, to ξ in (X,g) if for each � > 0, there is k0 ∈ N such that g(xjk − ξ) < � whenever

j,k ≥ k0. In symbols, one writes g2- lim x = ξ, and ξ is called the g2-limit of x.

Definition 2.2. Let (X,g) be a paranormed space. We say that a double sequence x = (xjk) is (λ,µ)-

statistically convergent, shortly, g(Sλ,µ)-convergent, to ξ in (X,g), if for each � > 0, the set {(j,k) ∈ N×N :

g(xjk − ξ) ≥ �} has (λ,µ)-density zero, equivalently, one writes

(P) lim
m,n

1

λmµn
|{(j,k),j ∈ Im,k ∈ Jn : g(xjk − ξ) ≥ �}| = 0.

In this case, one writes g(Sλ,µ)- lim x = ξ. If we choose λm = m and µn = n then the notion of g(Sλ,µ)-

convergence is reduced to statistically convergence for double sequence in (X,g) due to Arani et al. [5]. We

denote this by g(S2)-convergence and write g(S2)- lim x = ξ.

Theorem 2.3. If a double sequence x = (xjk) is g(Sλ,µ)-convergent then g(Sλ,µ)-limit is unique.

Proof. Assume that g(Sλ,µ)- lim x = ξ
′ and g(Sλ,µ)- lim x = ξ

′′. Let � > 0 be given. We now define the

following two sets B′(�) = {(j,k) ∈ N×N : g(xjk−ξ′) ≥ �/2} and B′′(�) = {(j,k) ∈ N×N : g(xjk−ξ′′) ≥ �/2}.



Int. J. Anal. Appl. 17 (5) (2019) 714

Since g(Sλ,µ)- lim x = ξ
′, one obtains

δλ,µ(B
′(�)) = (P) lim

m,n

1

λmµn
|{(j,k),j ∈ Im,k ∈ Jn : g(xjk − ξ′) ≥ �/2}| = 0.

Similarly, the assumption g(Sλ,µ)- lim x = ξ
′′ gives δλ,µ(B

′′(�)) = 0. Now, let B(�) = B′(�) ∪ B′′(�). Then

δλ,µ(B(�)) = 0 and so δλ,µ(B
c(�)) = 1 since Bc(�) is a nonempty set. Now if (j,k) ∈ N×N\B(�), then

g(ξ′ − ξ′′) ≤ g(xjk − ξ′) + g(xjk − ξ′′) < �/2 + �/2 = �.

Since � > 0 is arbitrary, one obtains g(ξ′ − ξ′′) = 0 which yields ξ′ = ξ′′. �

Theorem 2.4. If a double sequence x = (xjk) is g2-convergent to ξ, then it is g(Sλ,µ)-convergent to the

same limit.

Proof. Assume that (xjk) is g2-convergent to ξ, that is, g2- lim x = ξ. Let � > 0 be given. Then, there is

N ∈ N such that g(xjk − ξ) < � for all j,k ≥ N. Since, the set K(�) = {(j,k) ∈ N × N : g(xjk − ξ) ≥ �} is

contained in N×N, hence δλ,µ(K(�)) = 0, that is, (xjk) is g(Sλ,µ)-convergent to ξ.

Example 2.5. The present example proves that the converse of last Theorem is not true in general. Let

X = `(1/jk) =


x = (xjk) :

∞∑
j=1

∞∑
k=1

| xjk |1/jk< ∞




with the paranorm

g(x) =

∞∑
j=1

∞∑
k=1

| xjk |1/jk .

Let us define x = (xjk) by

xjk =


 jk for m− [

√
λm] + 1 ≤ j ≤ m,n− [

√
µn] + 1 ≤ k ≤ n,

0 otherwise.

For 0 < � < 1, one writes

B(�) := {(j,k) ∈ N×N : g(xjk) ≥ �}.

It is easy to see that

g(xjk) =


 (jk)

1/jk for m− [
√
λm] + 1 ≤ j ≤ m,n− [

√
µn] + 1 ≤ k ≤ n,

0 otherwise;

and hence we obtain

(P) lim
jk
g(xjk) :=


 1 for m− [

√
λm] + 1 ≤ j ≤ m,n− [

√
µn] + 1 ≤ k ≤ n,

0 otherwise.



Int. J. Anal. Appl. 17 (5) (2019) 715

We obtain that (xjk) is not convergent in (X,g) (g2- lim x does not exist) but δλ,µ(B(�)) = 0, that is,

g(Sλ,µ)- lim x = 0. Hereby, we conclude that the converse of above Theorem 2.4 need not be true in general.

�

The proof of the following theorem is straightforward and hence omitted.

Theorem 2.6. Let (X,g) be a paranormed space and assume that g(Sλ,µ)- lim x
′ = ξ′ and g(Sλ,µ)- lim x

′′ =

ξ′′. Then

(i) g(Sλ,µ)- lim(x
′ ±x′′) = ξ′ ± ξ′′,

(ii) g(Sλ,µ)- lim αx
′ = cξ′ for c ∈ R.

Theorem 2.7. Let (X,g) be a paranormed space. Then, a double sequence x = (xjk) is (λ,µ)-statistically

convergent to ξ in (X,g) if and only if there exists a set B = {(jm,kn) : j1 < j2 < ... < jm < ...; k1 < k2 <

... < kn < ...}⊆ N×N with δ(B) = 1 such that g- limm,n xjmkn = ξ.

Proof. Suppose that x = (xjk) is g(Sλ,µ)-convergent to ξ, that is, g(Sλ,µ)- lim x = ξ. For s = 1, 2, ..., one

writes

B(s) =

{
(m,n) ∈ N×N : g(xjmkn − ξ) ≥

1

s

}
and

D(s) =

{
(m,n) ∈ N×N : g(xjmkn − ξ) <

1

s

}
.

Then δλ,µ(B(s)) = 0,

D(1) ⊃ D(2) ⊃ ... ⊃ D(i) ⊃ D(i + 1) ⊃ ..., (1)

and

δλ,µ(D(s)) = 1 (s = 1, 2, ...). (2)

We need to prove that (xjmkn ) is g2-convergent to ξ for (m,n) ∈ D(s). Let us assume, on contrary, that

(xjmkn ) is not g2-convergent to ξ. Consequently, there is � > 0 such that g(xkn − ξ) ≥ � for infinitely many

terms. Let us write

D(�) = {(m,n) ∈ N×N : g(xjmkn − ξ) < �}

and � > 1
s
, s ∈ N. Then

δλ,µ(D(�)) = 0,

and by (1), D(s) ⊂ D(�). Hence δλ,µ(D(s)) = 0, which contradicts (2) and therefore (xjmkn ) is g2-convergent

to ξ.



Int. J. Anal. Appl. 17 (5) (2019) 716

Conversely, let us assume there exists a set B = {(jm,kn) : j1 < j2 < ... < jm < ...; k1 < k2 < k3 <

... < kn < ...} ⊆ N × N with δ(B) = 1 such that g- limn→∞xjmkn = ξ. Then there is a positive integer N

such that g(xmn − ξ) < � for m,n > N. Put

B(�) = {(m,n) ∈ N×N : g(xmn − ξ) ≥ �}

and B′ = {(jN+1,kN+1), (jN+2,kN+2), ...}. Then δλ,µ(B′) = 1 and B(�) ⊆ N × N −B′ which implies that

δλ,µ(B(�)) = 0. Hence g(Sλ,µ)- lim x = ξ. �

We are now defining the notion of (λ,µ)-statistically Cauchy double sequence in a paranormed space

and prove that it is equivalent to the notion of (λ,µ)-statistically convergence double sequence.

Definition 2.8. Let (X,g) be a paranormed space. We say that x = (xjk) is (λ,µ)-statistically Cauchy

double sequence in (X,g), denoted by g(Sλ,µ)-Cauchy, if for every � > 0 there exist M,N ∈ N such that, for

all j,m ≥ M, k,n ≥ N, we have

(P) lim
m,n

1

λmµn
|{(j,k),j ∈ Im,k ∈ Jn : g(xjk −xmn) ≥ �}| = 0.

Theorem 2.9. Let x = (xjk) be a double sequence in a complete paranormed space (X,g). Then, x is

g(Sλ,µ)-convergent iff it is g(Sλ,µ)-Cauchy.

Proof. Assume that g(Sλ,µ)- lim x = ξ. Then, the set

G(�) = {(j,k) ∈ N×N : g(xjk − ξ) ≥ �/2}

has (λ,µ)-density zero which yields

δλ,µ(G
c(�)) = δλ,µ({(j,k) ∈ N×N : g(xjk − ξ) < �}) = 1.

Suppose (m,n) ∈ Gc(�). Therefore, g(xmn − ξ) < �/2. Now, let

H(�) := {(j,k) ∈ N×N : g(xjk −xmn) ≥ �}.

We need to show that H(�) ⊂ G(�). Let (j,k) ∈ H(�). Then g(xjk −xmn) ≥ � and hence g(xjk − ξ) ≥ �/2,

i.e., (j,k) ∈ G(�). Otherwise, if g(xjk − ξ) < ε then

� ≤ g(xjk −xmn) ≤ g(xjk − ξ) + g(xmn − ξ) <
�

2
+
�

2
= �,

which is not possible. Thus H(�) ⊂ G(�) and hence

δλ,µ(H(�)) = δλ,µ ({(j,k) ∈ N×N : g(xjk −xmn) ≥ �}) = 0.

Therefore (xjk) is (λ,µ)-statistically Cauchy in (X,g).



Int. J. Anal. Appl. 17 (5) (2019) 717

Conversely, assume that x = (xjk) is g(Sλ,µ)-Cauchy but not g(Sλ,µ)-convergent. Then there exist

M,N ∈ N such that for all j,m ≥ M, k,n ≥ N, the set

A(�) = {(j,k) ∈ N×N : g(xjk −xmn) ≥ �},

has (λ,µ)-density zero, that is, δλ,µ(A(�) = 0 and δλ,µ(E(�)) = 0, where

E(�) =
{

(j,k) ∈ N×N : g(xjk − ξ) <
�

2

}
,

that is,

(P) lim
m,n

1

λmµn

∣∣∣{(j,k),j ∈ Im,k ∈ Jn : g(xjk − ξ) < �
2

}∣∣∣ = 0,
which yields δλ,µ(E

c(�)) = 1. Since g(xjk−xmn) ≤ 2g(xjk−ξ) < �, if g(xjk−ξ) < �/2. Thus, δλ,µ(AC(�)) = 0

and so δλ,µ(A(�) = 1. This is a contradiction to our assumption that x is g(Sλ,µ)-Cauchy. Hence x is g(Sλ,µ)-

convergent.

3. Strong summability for double sequences in (X,g)

We give the idea of strong (λ,µ)p-summability in the setting of paranormed space (X,g) and obtain

its relation with g(Sλ,µ)-convergence.

Definition 3.1. Let (X,g) be a paranormed space and let p be a positive real number. The double sequence

x = (xjk) is said to be strongly (λ,µ)p-summable to the limit ξ in (X,g), denoted by xjk −→ ξ[Vλ,µ,g]p, if

(P) lim
m,n

1

λmµn

∑
j∈Im

∑
k∈Jn

(g (xjk − ξ))
p

= 0 (0 < p < ∞).

Theorem 3.2. One has the following:

(i) If 0 < p < ∞ and xjk −→ ξ[Vλ,µ,g]p, then x = (xjk) is g(Sλ,µ)-convergent to ξ.

(ii) If x = (xjk) ∈L∞ and g(Sλ,µ)-convergent to ξ then xjk −→ ξ[Vλ,µ,g]p, where p ∈ (0,∞).

Proof. (i) Assume that xjk −→ ξ[Vλ,µ,g]p. Then, as m,n →∞, one obtains

0 ←−
1

λmµn

∑
j∈Im

∑
k∈Jn

(g(xjk − ξ))p ≥
1

λmµn

∑
j∈Im

(g(xk−ξ))p≥�

∑
k∈Jn

(g(xk−ξ))p≥�

(g(xk − ξ))p

≥
�p

λmµn
|F(�)|,

where F(�) = {j ∈ Im,k ∈ Jn : (g(xjk − ξ))p ≥ �}. That is, (P) limm,n→∞ 1λmµn |F(�)| = 0 and so

δλ,µ(F(�)) = (P) lim
m,n

1

λmµn
|{(j,k),j ∈ Im,k ∈ Jn : (g(xjk − ξ))p ≥ �}| = 0.

Thus x = (xjk) is g(Sλ,µ)-convergent to ξ.



Int. J. Anal. Appl. 17 (5) (2019) 718

(ii) Assume that a double sequence x = (xjk) is bounded and g(Sλ,µ)-convergent to ξ. Let � > 0 be

given. Then, we have δλ,µ(F(�)) = 0. Since x ∈L∞, there is an M > 0 such that g(xjk − ξ) ≤ M. We have

1

λmµn

∑
j∈Im

∑
k∈Jn

(g(xjk − ξ))p

=
1

λmµn

∑
j∈Im

(j,k)/∈F(�)

∑
k∈Jn

(j,k)/∈F(�)

(g(xjk − ξ))p +
1

λmµn

∑
j∈Im

(j,k)∈F(�)

∑
k∈Jn

(j,k)∈F(�)

(g(xjk − ξ))p.

If we take (j,k) /∈ F(�) then
1

λmµn

∑
j∈Im

(j,k)/∈F(�)

∑
k∈Jn

(j,k)/∈F(�)

(g(xjk − ξ))p < �.

On the other and, if (j,k) ∈ F(�), we have

1

λmµn

∑
j∈Im

(j,k)∈F(�)

∑
k∈Jn

(j,k)∈F(�)

(g(xjk − ξ))p

≤ (sup g(xjk − ξ))
1

λmµn
(|{j ∈ Im,k ∈ Jn : (g(xjk − ξ))p ≥ �}|)

≤
M

λmµn
|{j ∈ Im,k ∈ Jn : (g(xjk − ξ))p ≥ �}|.

We see that the right of above inequality tends to zero as m,n → ∞, since δλ,µ(F(�)) = 0. Hence, we

conclude that xk −→ ξ[Vλ,µ,g]p. �

Remark 3.3. If we choose λm = m and µn = n, then strong (λ,µ)p-summablity in a paranormed space is

reduced to the notion of strong p-Cesàro summablity for double sequences in the same setup, denoted by

[C1,1,g]p. Then, we have the following corollary from Theorem 3.3.

Corollary 3.4. Let (X,g) be a paranormed space.

(i) If p ∈ (0,∞) and xjk −→ ξ[C1,1,g]p, then g(S2)- lim x = ξ.

(ii) If x = (xjk) ∈L∞ and g(S2)- lim x = ξ then xjk −→ ξ[C1,1,g]p (p ∈ (0,∞)).

Theorem 3.5. If a double x = (xjk) is strongly (λ,µ)p-summable or (λ,µ)-statistically convergent to ξ in

(X,g), then there is a convergent double sequence y = (yjk) and a (λ,µ)-statistically null double sequence

z = (zjk) such that y = (yjk) is convergent to ξ in Pringsheim’s sense, x = y + z and

(P) lim
m,n

1

λmµn
|{(j,k),j ∈ Im,k ∈ Jn : zjk 6= 0}| = 0. (3)

Moreover, if a double sequence x = (xjk) is bounded, then both (yjk) and (zjk) are bounded.



Int. J. Anal. Appl. 17 (5) (2019) 719

Proof. It is clear from Theorem 3.2 that if a double sequence xjk −→ ξ[Vλ,µ,g]p, then it is (λ,µ)-statistically

convergent to ξ. Let us take S(0) = 0 and choose a strictly increasing sequence S(1) < S(2) < S(3) < ... of

positive integers such that

1

λmµn
|{(j,k),j ∈ Im,k ∈ Jn : |xjk − ξ| ≥ l}| < l−1,

for m,n > S(l). We are defining y = (yjk) and z = (zjk) as follows: Choose zjk = 0 and yjk = xjk if

S(0) < j,k < S(1). Suppose l ≥ 1 and S(l) < j,k < S(l + 1). We now set

yjk =


 xjk, zjk = 0 for |xjk − ξ| < l

−1,

ξ, zjk = xjk − ξ for |xjk − ξ| ≥ l−1.

Clearly, x = y + z and double sequences y and z are bounded if a double sequence x is bounded. We have to

show that y = (yjk) is convergent to ξ in the Pringsheim’s sense. For given � > 0, let us choose l such that

� > 1/l. We can see that for j,k > S(l), one obtains

|yjk − ξ| < � (since |yjk − ξ| = |xjk − ξ| < �) if |xjk − ξ| < l−1

and

|yjk − ξ| = |ξ − ξ| = 0 if |xjk − ξ| > l−1.

If follows that (yjk) is convergent to ξ in the Pringsheim’s sense. It remains to prove that (3) holds. It is

enough to prove that if δ > 0 and l ∈ N such that 1/l < δ, then

|{(j,k),j ∈ Im,k ∈ Jn : zjk 6= 0}| < δ ∀ m,n > S(l).

As we have seen from the construction that if S(l) < j,k ≤ S(l + 1) then zjk = 0 only if |xjk − ξ| > 1/l. It

follows that if S(r) < j,k ≤ S(r + 1), then

{(j,k),j ∈ Im,k ∈ Jn : zjk 6= 0}⊆{(j,k),j ∈ Im,k ∈ Jn : |xjk − ξ| > 1/r}.

Consequently, if S(r) < j,k ≤ S(r + 1) and r > l, one obtains

1

λmµn
|{(j,k),j ∈ Im,k ∈ Jn : zjk 6= 0}|

⊆
1

λmµn
|{(j,k),j ∈ Im,k ∈ Jn : |xjk − ξ| > 1/r}| < 1/r < 1/l < δ.

Thus, we have the following

(P) lim
m,n

1

λmµn
|{(j,k),j ∈ Im,k ∈ Jn : zjk 6= 0}| = 0.



Int. J. Anal. Appl. 17 (5) (2019) 720

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	1. Introduction and preliminaries
	2. (,)-Statistical convergence in paranormed spaces
	3. Strong summability for double sequences in (X,g)
	References