Statistical Convergence of Double Sequences of Order 𝛼

Abstract

We intend to make a new approach and introduce the concepts of statistical convergence of order 𝛼 and strongly 𝑝-Cesàro summability of order 𝛼 for double sequences of complex or real numbers. Also, some relations between the statistical convergence of order 𝛼 and strong 𝑝-Cesàro summability of order 𝛼 are given.

1. Introduction

The concept of statistical convergence was introduced by Steinhaus [1] and Fast [2] and later investigated by Schoenberg [3] independently for real and complex sequences. Over the years and under different names, statistical convergence has been discussed in the theory of Fourier analysis, ergodic theory, number theory, measure theory, trigonometric series, turnpike theory, and Banach spaces. Later on, it was further investigated from the sequence space point of view and linked with summability theory by Fridy [4], Connor [5], Maddox [6], Rath and Tripathy [7], Šalát [8], Tripathy et al. [9–12], and many others.

The idea of statistical convergence was later extended to double sequences by Mursaleen and Edely [13], Karakaya et al. [14], Móricz [15], and Tripathy et al. [16–18]. More recent developments on double sequences can be found in [19–22].

In the present paper, we introduce and examine the concepts of statistically convergence of double sequences of order 𝛼 and strong 𝑝-Cesàro summability of order 𝛼 of double sequences of complex or real numbers, where 𝛼 denotes the pair (𝑠,𝑡), and 𝑠,𝑡 are any real numbers such that 𝑠,𝑡∈(0,1]. The order of statistical convergence of a single sequence of numbers was given in [23], and after that, statistical convergence of order 𝛼 and strong 𝑝-Cesàro summability of order 𝛼 were studied by Çolak in [24, 25].

In Section 2, we give a brief information about statistical convergence and strong 𝑝-Cesàro summability, and we define the concepts of statistical convergence of order 𝛼 and strong 𝑝-Cesàro summability of order 𝛼 and give some results. In Section 3, we give the main results and establish some inclusion relations between 𝑆𝛼² and 𝑤²_𝑝𝛼.

2. Definitions and Preliminaries

In this section, mainly we introduce and examine the concepts of the 𝛼-double density of a subset 𝐸 of ℕ×ℕ, statistical convergence of order 𝛼, and strong 𝑝-Cesàro summability of order 𝛼 of the double sequences of complex or real numbers for 𝛼∈(0,1].

𝑤² denotes the space of all double sequences. Let ℓ²_∞, 𝑐² and 𝑐²₀ be the linear spaces of bounded, convergent, and null sequences 𝑥=(𝑥_𝑗𝑘) with complex terms, respectively, normed by ‖𝑥‖_(∞,2)=sup_𝑗,𝑘|𝑥_𝑗𝑘|, where 𝑗,𝑘∈ℕ={1,2,…} the set of positive integers.

By the convergence of a double sequence, we mean the convergence in Pringsheim’s sense [26]. A double sequence 𝑥=(𝑥_𝑗𝑘) is said to be convergent in the Pringsheim sense if for every 𝜀>0 there exists 𝑁∈ℕ such that |𝑥_𝑗𝑘−𝐿|<𝜀 whenever 𝑗,𝑘≥𝑁; 𝐿 is called the Pringsheim limit of 𝑥.

A double sequence 𝑥=(𝑥_𝑗𝑘) is bounded if there exists a positive number 𝑀 such that |𝑥_𝑗𝑘|≤𝑀 for all 𝑗 and 𝑘, that is, if ‖𝑥‖_(∞,2)=sup_𝑗,𝑘𝑥_𝑗𝑘<∞. (1) Note that in contrast to the case for single sequences, a convergent double sequence need not be bounded.

Let 𝐾⊆ℕ×ℕ and let 𝐾(𝑛,𝑚) be the number of (𝑖,𝑗) in 𝐾 such that 𝑖≤𝑛 and 𝑗≤𝑚. Then, the lower asymptotic density of a set 𝐾⊆ℕ×ℕ is defined by 𝛿²(𝐾)=liminf_𝑛,𝑚𝐾(𝑛,𝑚).𝑛𝑚 (2) In case the sequence (𝐾(𝑛,𝑚)/𝑛𝑚) has a limit in Pringsheim’s sense, then we say that 𝐾 has a double natural density and is defined by 𝛿²(𝐾)=lim_𝑛,𝑚𝐾(𝑛,𝑚).𝑛𝑚 (3)

Throughout the paper, we take 𝑠,𝑡,𝑢,𝑣∈(0,1], otherwise indicated, and for the sake of brevity, we write 𝛼 instead of (𝑠,𝑡) and 𝛽 instead of (𝑢,𝑣). Also, we define   𝛼⪯𝛽⇔𝑠≤𝑢 and 𝑡≤𝑣 𝛼≺𝛽⇔𝑠<𝑢 and 𝑡<𝑣 𝛼≅𝛽⇔𝑠=𝑢 and 𝑡=𝑣 𝛼∈(0,1]⇔𝑠,𝑡∈(0,1] 𝛽∈(0,1]⇔𝑢,𝑣∈(0,1] 𝛼≅1 in case 𝑠=𝑡=1 𝛽≅1 in case 𝑢=𝑣=1 𝛼≻1 in case 𝑠>1 and 𝑡>1,and furthermore, we write 𝑆𝛼² to denote 𝑆²_(𝑠,𝑡) and 𝑆²𝛽 to denote 𝑆²_(𝑢,𝑣) in the following.

Now we define the 𝛼-double density of the set 𝐾⊆ℕ×ℕ as 𝛿𝛼²(𝐾)=lim_𝑛,𝑚𝐾(𝑛,𝑚)𝑛^𝑠𝑚^𝑡. (4)

Remark 1. Note that for any set 𝐾⊆ℕ×ℕ, 𝛿𝛼²(𝐾) may be greater than 1, even equal to ∞, but 𝛿²(𝐾)≤1. Also, 𝛿²(𝐾^𝑐)=1−𝛿²(𝐾) holds but 𝛿𝛼²(𝐾)=1−𝛿𝛼²(𝐾) does not hold in general.

Definition 2. Let 𝑥=(𝑥_𝑗𝑘)∈𝑤² and 𝛼∈(0,1] be given. The sequence (𝑥_𝑗𝑘) is said to be statistically convergent of order 𝛼 if there is a complex number ℓ such that for every 𝜀>0, lim_𝑛→∞1𝑛^𝑠𝑚^𝑡𝑗,𝑘∶𝑗≤𝑛,𝑘≤𝑚∶𝑥_𝑗𝑘−ℓ≥𝜀=0, (5) which is the case when we say that 𝑥 is statistically convergent of order 𝛼 to ℓ. In this case, we write 𝑆𝛼²−lim_𝑗,𝑘𝑥_𝑗𝑘=ℓ and we denote the set of all statistically convergent double sequences of order 𝛼 by 𝑆𝛼².

In case 𝛼≅1, the statistical convergence of order 𝛼 reduces to the statistical convergence of double sequences [13]. If 𝑥=(𝑥_𝑗𝑘) is statistically convergent of order 𝛼 to the number ℓ, then ℓ is determined uniquely. The statistical convergence of order 𝛼 is well defined for 𝛼∈(0,1], but it is not well defined for 𝛼≻1. For this, let 𝑥=(𝑥_𝑗𝑘) be defined as follows: 𝑥_𝑗𝑘=1,if𝑗+𝑘even0,if𝑗+𝑘odd. (6) Then, lim_{𝑛,𝑚→∞}1𝑛^𝑠𝑚^𝑡𝑗,𝑘∶𝑗≤𝑛,𝑘≤𝑚,𝑥_𝑗𝑘−1≥𝜀≤lim_{𝑛,𝑚→∞}(𝑛/2+1)(𝑚/2+1)𝑛^𝑠𝑚^𝑡=0,lim_{𝑛,𝑚→∞}1𝑛^𝑠𝑚^𝑡𝑗,𝑘∶𝑗≤𝑛,𝑘≤𝑚,𝑥_𝑗𝑘≥𝜀≤lim_{𝑛,𝑚→∞}(𝑛/2+1)(𝑚/2+1)𝑛^𝑠𝑚^𝑡=0 (7) for 𝛼≻1 that is 𝑠>1 and 𝑡>1, so that 𝑥=(𝑥_𝑗𝑘) statistically converges of order 𝛼 both to 1 and 0, that is, 𝑆𝛼²−lim𝑥_𝑗𝑘=1 and 𝑆𝛼²−lim𝑥_𝑗𝑘=0. However, this is impossible.

Theorem 3. Let 𝛼∈(0,1] and 𝑥=(𝑥_𝑗𝑘), 𝑦=(𝑦_𝑗𝑘) be sequences of complex numbers. Then,(i)If 𝑆𝛼²−lim𝑥_𝑗𝑘=𝑥₀ and 𝑐∈ℂ, then 𝑆𝛼²−lim𝑐𝑥_𝑗𝑘=𝑐𝑥₀(ii)If 𝑆𝛼²−lim𝑥_𝑗𝑘=𝑥_𝑜 and 𝑆𝛼²−lim𝑦_𝑗𝑘=𝑦₀, then 𝑆𝛼²−lim(𝑥_𝑗𝑘+𝑦_𝑗𝑘)=𝑥₀+𝑦₀.

Proof. (i) is clear in case 𝑐=0. Suppose that 𝑐≠0; then the proof of (i) follows from 1𝑛^𝑠𝑚^𝑡𝑗,𝑘∶𝑗≤𝑛,𝑘≤𝑚,𝑐𝑥_𝑗𝑘−𝑐𝑥₀=1≥𝜀𝑛^𝑠𝑚^𝑡𝑗,𝑘∶𝑗≤𝑛,𝑘≤𝑚,𝑥_𝑗𝑘−𝑥₀≥𝜀|𝑐| (8) and that of (ii) follows from the following inequality: 1𝑛^𝑠𝑚^𝑡𝑥𝑗,𝑘∶𝑗≤𝑛,𝑘≤𝑚,_𝑗𝑘+𝑦_𝑗𝑘−𝑥₀+𝑦₀≤1≥𝜀𝑛^𝑠𝑚^𝑡𝑗,𝑘∶𝑗≤𝑛,𝑘≤𝑚,𝑥_𝑗𝑘−𝑥₀≥𝜀2+1𝑛^𝑠𝑚^𝑡𝑗,𝑘∶𝑗≤𝑛,𝑘≤𝑚,𝑦_𝑗𝑘−𝑦₀≥𝜀2. (9)

It is easy to see that every convergent double sequence is statistically convergent of order 𝛼 to the same number, that is 𝑐²⊂𝑆𝛼² for each 𝛼∈(0,1] that is for each pair of (𝑠,𝑡) such that 𝑠,𝑡∈(0,1]. However, the converse does not hold. For example, the sequence 𝑥=(𝑥_𝑗𝑘) defined by 𝑥_𝑗𝑘=1,if𝑗=𝑛³,𝑘=𝑚³0,otherwise𝑛=1,2,,…;𝑚=1,2,… (10) is statistically convergent of order 𝛼 with 𝑆𝛼²−lim𝑥_𝑗𝑘=0 for 𝛼≻1/3 that is 𝑠>1/3, 𝑡>1/3 but it is not convergent.

Definition 4. Let 𝛼∈(0,1] be given. A sequence 𝑥=(𝑥_𝑗𝑘) is said to be Cesàro summable of order 𝛼 if there is a complex number ℓ such that lim_{𝑛,𝑚→∞}1𝑛^𝑠𝑚^𝑡𝑛𝑚_𝑗=1𝑘=1𝑥_𝑗𝑘=ℓ, (11) which is the case when we say that 𝑥 is Cesàro summable of order 𝛼 to ℓ. For 𝛼≅1, the Cesàro summability of order 𝛼 reduces to the Cesàro summability that is given in [15]. The set of all Cesàro summable double sequences of order 𝛼 will be denoted by 𝑤𝛼². The set of all Cesàro summable double sequences will be denoted by 𝑤².

Definition 5. Let 𝛼∈(0,1] be given, and let 𝑝 be a positive real number. Then, a sequence 𝑥=(𝑥_𝑗𝑘) is said to be strongly 𝑝-Cesàro summable of order 𝛼 if there is a complex number ℓ such that lim_{𝑛,𝑚→∞}1𝑛^𝑠𝑚^𝑡𝑛𝑚_𝑗=1𝑘=1𝑥_𝑗𝑘−ℓ^𝑝=0, (12) which is the case when we say that 𝑥 is strongly 𝑝-Cesàro summable of order 𝛼 to ℓ. For 𝛼≅1, the strong 𝑝-Cesàro summability of order 𝛼 reduces to the strong 𝑝-Cesàro summability that is given in [13]. The set of all strongly 𝑝-Cesàro summable sequences of order 𝛼 will be denoted by [𝑤²_𝑝]𝛼.

3. Main Results

In this section, we give the main results of the paper. In Theorem 6, we give the relationship between the statistical double convergence of order 𝛼 and the statistical double convergence of order 𝛽 for 𝛼⪯𝛽 and so that the relationship between the statistical double convergence of order 𝛼 and the statistical convergence. In Corollary 10, we give the relationship between the strong 𝑝-Cesàro summability of order 𝛼 and the strong 𝑝-Cesàro summability of order 𝛽. In Theorem 12, we give the relationship between the strong 𝑝-Cesàro summability of order 𝛼 and the statistical double convergence of order 𝛽.

Theorem 6. Let 𝛼,𝛽∈(0,1] be given such that 𝛼⪯𝛽. Then, 𝑆𝛼²⊆𝑆²𝛽 and the inclusion is strict for some 𝛼 and 𝛽 such that 𝛼≺𝛽.

Proof. Let 𝛼,𝛽∈(0,1] be given. If 𝛼⪯𝛽 and so that 𝑠≤𝑢 and 𝑡≤𝑣, then 1𝑛^𝑢𝑚^𝑣𝑗,𝑘∶𝑗≤𝑛,𝑘≤𝑚,𝑥_𝑗𝑘≤1−ℓ≥𝜀𝑛^𝑠𝑚^𝑡𝑗,𝑘∶𝑗≤𝑛,𝑘≤𝑚,𝑥_𝑗𝑘−ℓ≥𝜀 (13) for every 𝜀>0 and this gives that 𝑆𝛼²⊆𝑆²𝛽. To show that the inclusion is strict, consider the sequence 𝑥=(𝑥_𝑗𝑘) defined by 𝑥_𝑗𝑘=1,𝑗=𝑛²,𝑘=𝑚²0,otherwise𝑚,𝑛=1,2,3,… (14) Then, 𝑆²𝛽−lim𝑥_𝑘=0; that is, 𝑥∈𝑆²𝛽 for 𝛽∈(1/2,1] (i.e., for 1/2<𝑢≤1 and 1/2<𝑣≤1) but 𝑥∉𝑆𝛼² for 𝛼∈(0,1/2] (i.e., for 0<𝑠≤1/2 and 0<𝑡≤1/2).

If we take 𝛽≅1 in Theorem 6, then we obtain the following result.

Corollary 7. If a double sequence is statistically convergent of order 𝛼 to ℓ, for some 𝛼 such that 𝛼∈(0,1], then it is statistically convergent to ℓ; that is 𝑆𝛼²⊆𝑆², and the inclusion is strict.

From Theorem 6, we have the following results and the proof is easy.

Corollary 8. Let 𝛼,𝛽∈(0,1] be given. Then,(i)𝑆𝛼²=𝑆²𝛽 if and only if 𝛼≅𝛽;(ii)𝑆𝛼²=𝑆² if and only if 𝛼≅1.

The proof of the following theorem is straight forward from the definition, so omitted.

Theorem 9. Let 𝛼,𝛽∈(0,1] be given such that 𝛼⪯𝛽, and let 𝑝 be a positive real number. Then, [𝑤²_𝑝]𝛼⊆[𝑤²_𝑝]𝛽 and the inclusion is strict for some 𝛼 and 𝛽 such that 𝛼≺𝛽.

Proof. Let 𝑥=(𝑥_𝑘)∈[𝑤²_𝑝]𝛼. Then, given 𝛼 and 𝛽 such that 𝛼⪯𝛽 and a positive real number 𝑝, we may write 1𝑛^𝑢𝑚^𝑣𝑛𝑚_𝑗=1𝑘=1𝑥_𝑗𝑘−ℓ^𝑝≤1𝑛^𝑠𝑚^𝑡𝑛𝑚_𝑗=1𝑘=1𝑥_𝑗𝑘−ℓ^𝑝, (15) and this gives that [𝑤²_𝑝]𝛼⊆[𝑤²_𝑝]𝛽.
To show that the inclusion is strict, consider the sequence 𝑥=(𝑥_𝑗𝑘) defined in (14). It is easy to see that 1𝑛^𝑢𝑚^𝑣𝑛𝑚_𝑗=1𝑘=1𝑥_𝑗𝑘^𝑝≤√𝑛√𝑚𝑛^𝑢𝑚^𝑣=1𝑛^𝑢−1/2𝑚^𝑣−1/2. (16) Since 1/(𝑛^𝑢−1/2𝑚^𝑣−1/2)→0 as 𝑛→∞, 𝑚→∞, then [𝑤²_𝑝]𝛽−lim𝑥_𝑗𝑘=0;thatis,𝑥∈[𝑤²_𝑝]𝛽 for 𝛽∈(1/2,1] (i.e., for 1/2<𝑢≤1 and 1/2<𝑣≤1), but since √𝑛−1𝑛^𝑠√𝑚−1𝑚^𝑡≤1𝑛^𝑠𝑚^𝑡𝑛𝑚_𝑗=1𝑘=1𝑥_𝑗𝑘^𝑝 (17) and ((√𝑛−1)/𝑛^𝑠)((√𝑚−1)/𝑚^𝑡)→∞ as 𝑛→∞, 𝑚→∞, then 𝑥∉[𝑤²_𝑝]𝛼 for 𝛼∈(0,1/2] (i.e., for 0<𝑠<1/2 and 0<𝑡<1/2). This completes the proof.

The following result is a consequence of Theorem 9.

Corollary 10. Let 𝛼,𝛽∈(0,1] be given such that 𝛼⪯𝛽, and let 𝑝 be a positive real number. Then,(i)[𝑤²_𝑝]𝛼=[𝑤²_𝑝]𝛽 if and only if 𝛼≅𝛽;(ii)[𝑤²_𝑝]𝛼⊆𝑤²_𝑝  for each 𝛼 such that 𝛼∈(0,1] and 0<𝑝<∞.

The following result is a simple consequence of Hölder’s inequality which is an extension of a result of Maddox [27].

Theorem 11. Let 𝛼∈(0,1], and let 0<𝑝<𝑞<∞. Then, [𝑤²_𝑞]𝛼⊂[𝑤²_𝑝]𝛼.

Taking 𝛼≅1 in Theorem 11, we obtain a result of Maddox [27]: if 0<𝑝<𝑞<∞, then 𝑤²_𝑞⊂𝑤²_𝑝.

Theorem 12. Let 𝛼 and 𝛽 be given such that 𝛼⪯𝛽, and let 0<𝑝<∞, where 𝛼,𝛽∈(0,1]. If a sequence is strongly 𝑝-Cesàro summable of order 𝛼 to ℓ, then it is statistically convergent of order 𝛽 to ℓ.

Proof. For any sequences 𝑥=(𝑥_𝑗𝑘) and 𝜀>0, we have ^𝑛𝑚_𝑗=1𝑘=1𝑥_𝑗𝑘−ℓ^𝑝≥𝑗,𝑘∶𝑗≤𝑛,𝑘≤𝑚,𝑥_𝑗𝑘−ℓ≥𝜀⋅𝜀^𝑝, (18) so that since 𝛼⪯𝛽, 1𝑛^𝑠𝑚^𝑡𝑛𝑚_𝑗=1𝑘=1𝑥_𝑗𝑘−ℓ^𝑝≥1𝑛^𝑠𝑚^𝑡𝑗,𝑘∶𝑗≤𝑛,𝑘≤𝑚,𝑥_𝑗𝑘−ℓ≥𝜀⋅𝜀^𝑝≥1𝑛^𝑢𝑚^𝑣𝑗,𝑘∶𝑗≤𝑛,𝑘≤𝑚∶𝑥_𝑗𝑘−ℓ≥𝜀⋅𝜀^𝑝. (19) From this, it follows that if 𝑥=(𝑥_𝑗𝑘) is strongly 𝑝-Cesàro summable of order 𝛼 to ℓ, then it is statistically convergent of order 𝛽 to ℓ.

If we take 𝛽≅𝛼 in Theorem 12, we obtain the following result.

Corollary 13. Let 𝛼∈(0,1] be given, and let 0<𝑝<∞. If a double sequence is strongly 𝑝-Cesàro summable of order 𝛼 to ℓ, then it is statistically convergent of order 𝛼 to ℓ.

Remark 14. Note that the converse of Theorem 12 does not hold in general. We see that a bounded and statistically convergent double sequence of order 𝛼 need not be strongly 𝑝-Cesàro summable of order 𝛼 in general for 𝛼∈(0,1].

The sequence 𝑥=(𝑥_𝑗𝑘) defined by 𝑥_𝑗𝑘=1𝑗√𝑘,𝑗≠𝑛³,𝑘≠𝑚³1,otherwise𝑚,𝑛=1,2,3,… (20) is an example for this case. It is clear that 𝑥∈ℓ²_∞ and 𝑥∈𝑆𝛼² for each 𝛼∈(1/3,1]. First, recall that the inequality ^𝑛𝑚_𝑗=1𝑘=11𝑗√𝑘>√𝑚√𝑛 (21) holds for every positive integer 𝑚≥2 and 𝑛≥2. Define 𝐺_𝑛={𝑗≤𝑛∶𝑗≠𝑖²,𝑖=1,2,3,…}, 𝐻_𝑚={𝑘≤𝑚∶𝑘≠𝑖²,𝑖=1,2,3,…} and take 𝑝=1. Since ^𝑛𝑚_𝑗=1𝑘=1𝑥_𝑗𝑘^𝑝=^𝑛𝑚_𝑗=1𝑘=1𝑥_𝑗𝑘=_{𝑗∈𝐺𝑛,1≤𝑗≤𝑛𝑘∈𝐻𝑚,1≤𝑘≤𝑚}𝑥_𝑗𝑘+_{𝑗∉𝐺𝑛,1≤𝑗≤𝑛𝑘∉𝐻𝑚,1≤𝑘≤𝑚}𝑥_𝑗𝑘=_{𝑗∈𝐺𝑛,1≤𝑗≤𝑛𝑘∈𝐻𝑚,1≤𝑘≤𝑚}1𝑗√𝑘+_{𝑗∉𝐺𝑛,1≤𝑗≤𝑛𝑘∉𝐻𝑚,1≤𝑘≤𝑚}1>^𝑛𝑚_𝑗=1𝑘=11𝑗√𝑘>√𝑚√𝑛, (22) we have 1𝑛^𝑠𝑚^𝑡𝑛𝑚_𝑗=1𝑘=1𝑥_𝑗𝑘^𝑝=1𝑛^𝑠𝑚^𝑡𝑛𝑚_𝑗=1𝑘=1𝑥_𝑗𝑘>1𝑛^𝑠𝑚^𝑡𝑛𝑚_𝑗=1𝑘=11𝑗√𝑘>1𝑛^𝑠𝑚^𝑡√𝑛√𝑚=1𝑛^𝑠−1/2⋅1𝑚^𝑡−1/2⟶∞as𝑚⟶∞,𝑛⟶∞, (23) so that 𝑥∉[𝑤²_𝑝]𝛼 for 𝛼∈(0,1/2] if 𝑝=1. Therefore, 𝑥∈𝑆𝛼²−[𝑤²_𝑝]𝛼 for 𝛼∈(1/3,1/2] if 𝑝=1.

Corollary 15. Let 𝛼∈(0,1], and let 𝑝 be a positive real number. Then, 𝑤²_𝑝𝛼⊂𝑆². The inclusion is strict if 𝛼∈(0,1).

Proof. From Corollaries 13 and 7, we have [𝑤²_𝑝]𝛼⊂𝑆². To show that the inclusion is strict, consider the sequence 𝑥=(𝑥_𝑗𝑘) defined in (10). Then, clearly 𝑆²−lim𝑥_𝑗𝑘=0; that is, 𝑥∈𝑆² but 𝑥∉[𝑤²_𝑝]𝛼 for 𝛼∈(0,1/3] and 𝑝=1. Indeed it is easy to see that 1𝑛^𝑠𝑚^𝑡𝑛𝑚_𝑗=1𝑘=1𝑥_𝑗𝑘−0^𝑝=1𝑛^𝑠𝑚^𝑡𝑛𝑚_𝑗=1𝑘=1𝑥_𝑗𝑘≥³√𝑛−1𝑛^𝑠⋅³√𝑚−1𝑚^𝑡. (24) Since (√³𝑛−1)/𝑛^𝑠→∞ as 𝑛→∞ and (√³𝑚−1)/𝑚^𝑡→∞ as 𝑚→∞, then 𝑥∉[𝑤²_𝑝]𝛼 if 𝛼∈(0,1/3] and 𝑝=1. Consequently, 𝑥∈𝑆²−[𝑤²_𝑝]𝛼 for 𝛼∈(0,1/3] and 𝑝=1. This completes the proof.

4. Conclusion

The concepts of statistical convergence and strong Cesàro summability of double sequences of complex or real numbers have been studied by various mathematicians. In this paper, we introduced the concepts of statistical convergence of order 𝛼 and strong 𝑝-Cesàro summability of order 𝛼 for double sequences, where 𝛼∈(0,1]. These concepts are much more general than the concepts of statistical convergence and strong 𝑝-Cesàro summability of double sequences that include these concepts in the special case 𝛼≅1.

Note that the converse of Theorem 12 does not hold in general; that is, a statistically convergent sequence of order 𝛼 (even a bounded and statistically convergent sequence of order 𝛼) need not be strongly 𝑝-Cesàro summable of order 𝛼 in general.