Some Tauberian Theorems under Triple Statistically Nörlund-Cesáro Summability Method

Abstract

In this paper, we extend the notion presented by Braha (2020) in a higher dimension, we introduce the notion of Np,qn,m,gCn,m,g1,1,1-statistically convergence and show necessity and sufficiency conditions under which the existence of the limit st-limn,m,g→∞xn,m,g=L follows from that st-limn,m,g→∞Np,qn,m,gCn,m,g1,1,1=L. These conditions are one-sided or two-sided if xn,m,g is a sequence of real or complex numbers, respectively.

1. Introduction

The concept of statistical convergence was introduced by Fast [1] and Steinhaus [2]. Besides, in this connection, Fridy [3] showed some relation to a Tauberian condition for the statistical convergence of $\left(x_k\right)$. Subsequently, many researchers have worked in this area in several settings. For more recent works in this direction, one may refer to [4, 5]. Existing works in this field based on statistical convergence appears to have been restricted to real or complex sequences; however, Parida et al. [6] extended the idea for a locally convex Hausdorff topological linear space. Tauber [7] introduced the first Tauberian theorems for single sequences, that an Abel summable sequence is convergent with some suitable conditions. Later, a huge number of authors such as Landau [8], Hardy and Littlewood [9], and Schmidt [10] obtained some classical Tauberian theorems for Cessáro and Abel summability methods of single sequences. Recently, Braha [11] introduced some notions on statistical convergence by using the Nörlund-Cesáro summability method in a single sequence and proved some Tauberian theorems. In the last year, Canak and Totur [12], and Jena et al. [13] presented and studied several Tauberian theorems for single sequences. On the other hand, Knopp [14] obtained some classical type Tauberian theorems for Abel and $\left(C\mathrm{,1,1}\right)$ summability methods of double sequences and proved that Abel and $\left(C\mathrm{,1,1}\right)$ summability methods hold for the set of bounded sequences. Further, Moricz [15] proved some Tauberian theorems for Cesáro summable double sequences and deduced Tauberian theorems of Landau [16] and Hardy [17] type. Canak and Totur [18] have proved a Tauberian theorem for Cesáro summability of single integrals and also the alternative proofs of some classical type Tauberian theorems for the Cesáro summability of single integrals and later introduced by Parida et al. [6] for double integrals. Otherwise, the notion of $\left(C\mathrm{,1,1,1}\right)$ summability of a triple sequence was originally introduced by Canak and Totur in 2016 [19]. Later, Canak et al. [20] studied some $\left(C\mathrm{,1,1,1}\right)$ means of a statistical convergent triple sequence and gave some classical Tauberian theorems for a triple sequence that $P$-convergence follows from statistically $\left(C\mathrm{,1,1,1}\right)$ summability under the two-sided boundedness conditions and slowly oscillating conditions in certain senses. Then, in 2020 Totur and Canak [21] proved Tauberian conditions under which convergence of triple integrals follows from $\left(C\mathrm{,1,1,1}\right)$ summability. For more studies associated to the main topic of this paper, we refer the reader to [22, 23, 24].

Let $\left(p_{n,m,g}\right)$ and $\left(q_{n,m,g}\right)$ be any two non-negative real sequences with

$$R_{n,m,g}=\sum _{i=0}^n\sum _{j=0}^m\sum _{k=0}^g{p}_{i,j,k}{q}_{n-i,m-j,g-k}\ne 0\left(\left(n,m,g\right)\in \mathrm{\mathbb{N}}\times \mathrm{\mathbb{N}}\times \mathrm{\mathbb{N}}\right)$$

and $\left(C\mathrm{,1,1,1}\right)$-Cesáro summability method. Let $\left(x_{n,m,g}\right)$ be a sequence of real of complex numbers and set

$$N_{p,q}^{n,m,g}{C}_{n,m,g}^{\left(\mathrm{1,1,1}\right)}=\frac{1}{R_{n,m,g}}\sum _{i=0}^n\sum _{j=0}^m\sum _{k=0}^g{p}_{i,j,k}{q}_{n-i,m-j,g-k}\frac{1}{i+1}\frac{1}{j+1}\frac{1}{k+1}\sum _{u=0}^i\sum _{v=0}^j\sum _{y=0}^k{x}_{u,v,y}$$

for $\left(n,m,g\right)\in \mathrm{\mathbb{N}}\times \mathrm{\mathbb{N}}\times \mathrm{\mathbb{N}}$.

In this paper, we show necessary and sufficient conditions under which the existence of the limit $\underset{n,m,g\to \infty }{lim}{x}_{n,m,g}=L$ follows from that of $\underset{n,m,g\to \infty }{lim}{N}_{p,q}^{n,m,g}{C}_{n,m,g}^{\left(\mathrm{1,1,1}\right)}=L$. These conditions are one-sided or two-sided if $\left(x_{n,m,g}\right)$ is a sequence of real or complex numbers, respectively.

Given two non-negative sequences $\left(p_{n,m,g}\right)$ and $\left(q_{n,m,g}\right)$, the convolution $\left(p\star q\right)$ is defined by

$$R_{n,m,g}={\left(p\star q\right)}_{n,m,g}=\sum _{i=0}^n\sum _{j=0}^m\sum _{k=0}^g{p}_{i,j,k}{q}_{n-i,m-j,g-k}=\sum _{i=0}^n\sum _{j=0}^m\sum _{k=0}^g{p}_{n-i,m-j,g-k}{q}_{i,j,k}$$

with $\left(C\mathrm{,1,1,1}\right)$ we will denote the triple Cesáro summability method. Now, let $\left(x_{n,m,g}\right)$ be a sequence, when ${\left(p\star q\right)}_{n,m,g}\ne 0$ for all $\left(n,m,g\right)\in \mathrm{\mathbb{N}}\times \mathrm{\mathbb{N}}\times \mathrm{\mathbb{N}}$ the generalized Nörlund-Cesáro transform of the sequence $\left(x_{n,m,g}\right)$ is the sequence $N_{p,q}^{n,m,g}{C}_{n,m,g}^{\left(\mathrm{1,1,1}\right)}$ obtained by putting

$$N_{p,q}^{n,m,g}{C}_{n,m,g}^{\left(\mathrm{1,1,1}\right)}=\frac{1}{{\left(p\star q\right)}_{n,m,g}}\sum _{i=0}^n\sum _{j=0}^m\sum _{k=0}^g{p}_{i,j,k}{q}_{n-i,m-j,g-k}\frac{1}{i+1}\frac{1}{j+1}\frac{1}{k+1}\sum _{u=0}^i\sum _{v=0}^j\sum _{y=0}^k{x}_{u,v,y}.$$E1

We say that the sequence $\left(x_{n,m,g}\right)$ is generalized Nörlund-Cesáro summable to $L$ determined by the sequences $\left(p_{n,m,g}\right)$ and $\left(q_{n,m,g}\right)$ (or simply summable $N_{p,q}^{n,m,g}{C}_{n,m,g}^{\left(\mathrm{1,1,1}\right)}$) to $L$ if

$$\underset{n,m,g\to \infty }{lim}{N}_{p,q}^{n,m,g}{C}_{n,m,g}^{\left(\mathrm{1,1,1}\right)}=L.$$E2

Throughout this paper, we will assume that the sequences $\left(p_{n,m,g}\right)$ and $\left(q_{n,m,g}\right)$ are satisfying the following conditions

$$q_{n,m,g}\ge 1,\sum _{i=0}^n\sum _{j=0}^m\sum _{k=0}^g{p}_{i,j}\sim \mathit{nmg},\left(n,m,g\right)\in \mathrm{\mathbb{N}}\times \mathrm{\mathbb{N}}\times \mathrm{\mathbb{N}},$$E3

$$q_{{\lambda }_{n-i,m-j,g-k}}\le 2q_{n-i,m-j,g-k},i=1,2,\dots ,n;j=1,2,\dots ,m;\lambda >1k=1,2,\dots ;\lambda >1,$$E4

$$q_{n-i,m-j,g-k}\le 2q_{{\lambda }_{n-i,m-j,g-k}}i=1,2,\dots ,{\lambda }_n;j=1,2,\dots ,{\lambda }_m;k=1,2,\dots ,;0<\lambda <1,$$E5

where ${\lambda }_n=\left[\mathit{\lambda n}\right]$, ${\lambda }_m=\left[\mathit{\lambda m}\right]$ and ${\lambda }_g=\left[\mathit{\lambda g}\right]$. On the other hand, $a_{n,m,g}\sim b_{n,m,g}$ means that there are constants $C,C_1$ such that $a_{n,m,g}\le {\mathit{Cb}}_{n,m,g}\le C_1{a}_{n,m,g}$. If

$$\underset{n,m,g\to \infty }{lim}{x}_{n,m,g}=L$$E6

implies (2), then the method $N_{p,q}^{n,m,g}{C}_{n,m,g}^{\left(\mathrm{1,1,1}\right)}$ is said to be regular. Nevertheless, the converse is not always true as can be seen in the following example:

Let us consider that $p_{n,m,g}=q_{n,m,g}=1$ for all $\left(n,m,g\right)\in \mathrm{\mathbb{N}}\times \mathrm{\mathbb{N}}\times \mathrm{\mathbb{N}}$. Besides, we define the following sequence $x=\left(x_{i,j,k}\right)={\left(-1\right)}^{i+j+k}$, then we get

$$\begin{array}{l}\frac{1}{\left(n+1\right)\left(m+1\right)\left(g+1\right)}\mid \sum _{i=0}^n\sum _{j=0}^m\sum _{k=0}^g\frac{1}{\left(i+1\right)\left(j+1\right)\left(k+1\right)}\sum _{u=0}^i\sum _{v=0}^j\sum _{y=0}^k{\left(-1\right)}^{u+v+y}\mid \\ \le \frac{1}{\left(n+1\right)\left(m+1\right)\left(g+1\right)}\sum _{i=0}^n\sum _{j=0}^m\sum _{k=0}^g\frac{1}{\left(i+1\right)\left(j+1\right)\left(k+1\right)}\sum _{u=0}^i\sum _{v=0}^j\sum _{y=0}^{k}1\to 1\mathit{as}n,m,g\to \infty .\end{array}$$

and as we know, $x=\left(x_{i,j,k}\right)$ is not convergent. Notice that (6) can imply (2) under a certain condition, which is called a Tauberian conditions. Any theorem which states that convergence of a sequence follows from its $N_{p,q}^{n,m,g}{C}_{n,m,g}^{\left(\mathrm{1,1,1}\right)}$ summability and some Tauberian conditions are said to be a Tauberian theorems for the $N_{p,q}^{n,m,g}{C}_{n,m,g}^{\left(\mathrm{1,1,1}\right)}$ summability method.

Next, we will find some conditions under which the converse implication holds, for defined convergence. Exactly, we will prove under which conditions statistical convergence of sequences $\left(x_{n,m,g}\right)$, follows from statistically Nörlund-Cesáro summability method.

A sequence $\left(x_{n,m,g}\right)$ is weighted $N_{p,q}^{n,m,g}{C}_{n,m,g}^{\left(\mathrm{1,1,1}\right)}$-statistically convergent to $L$ if for every $\epsilon >0$,

$$\begin{array}{l}\underset{n,m,g\to \infty }{lim}\frac{1}{{\left(p\star q\right)}_{n,m,g}}\mid \{i,j,k\le {\left(p\star q\right)}_{n,m,g}:\frac{1}{{\left(p\star q\right)}_{n,m,g}}\sum _{i=0}^n\sum _{j=0}^m\sum _{k=0}^g{p}_{i,j,k}{q}_{n-i,m-j,g-k}\\ \frac{1}{i+1}\frac{1}{j+1}\frac{1}{\left(k+1\right)}\sum _{u=0}^i\sum _{v=0}^j\sum _{y=0}^k{x}_{u,v,y}-L\mid \ge \epsilon \}\mid =0.\end{array}$$

And we say that the sequence $\left(x_{n,m,g}\right)$ is statistically summable to $L$ by the weighted summability method $N_{p,q}^{n,m,g}{C}_{n,m,g}^{\left(\mathrm{1,1,1}\right)}$ if $\mathit{st}-\underset{n,m,g}{lim}{N}_{p,q}^{n,m,g}{C}_{n,m,g}^{\left(\mathrm{1,1,1}\right)}=L$. We will denote by $N_{p,q}^{n,m,g}{C}_{n,m,g}^{\left(\mathrm{1,1,1}\right)}\left(\mathit{st}\right)$ the set of all sequences which are statistically summable, by the weighted summability method $N_{p,q}^{n,m,g}{C}_{n,m,g}^{\left(\mathrm{1,1,1}\right)}$.

Theorem 1.1 Let $x=\left(x_{n,m,g}\right)$ be a sequence $N_{p,q}^{n,m,g}{C}_{n,m,g}^{\left(\mathrm{1,1,1}\right)}$ summable to $L$, then the sequence $x=\left(x_{n,m,g}\right)$ is $N_{p,q}^{n,m,g}{C}_{n,m,g}^{\left(\mathrm{1,1,1}\right)}$-statistically convergent to $L$, but not conversely.

Proof: The first part of the proof is obvious. To prove the second part, we will show the following example:

Let us define

$$x_{i,j,k}=\left\{\begin{array}{ll}\sqrt{\mathit{xyz}},& \mathit{\text{for}}i=n^{2}j=m^2\mathit{\text{and}}k=g^2\\ 0,& \mathit{\text{otherwise}}\end{array}\right.$$

and $p_{n,m,g}=1=q_{n,m,g}$. Under this conditions we obtain,

$$\begin{array}{l}\frac{1}{\left(n+1\right)\left(m+1\right)\left(g+1\right)}\mid \{i,j,k\le n+1,m+1,g+1:\\ \mid \frac{1}{\left(n+1\right)\left(m+1\right)\left(g+1\right)}\sum _{i=0}^n\sum _{j=0}^m\sum _{k=0}^g\frac{1}{P_{i,j,k}}\sum _{u=0}^i\sum _{v=0}^j\sum _{y=0}^k{p}_{u,v,y}{x}_{u,v,y}-0\mid \ge \epsilon \}\mid \\ \le \frac{\sqrt{\left(n+1\right)\left(m+1\right)\left(g+1\right)}}{\left(n+1\right)\left(m+1\right)\left(g+1\right)}\to 0.\end{array}$$

On the other hand, for $i=n^2$, $j=m^2$ and $k=g2$, we have

$$\begin{array}{l}\frac{1}{\left(n+1\right)\left(m+1\right)\left(g+1\right)}\sum _{i=0}^n\sum _{j=0}^m\sum _{k=0}^g\frac{1}{\left(i+1\right)\left(j+1\right)\left(k+1\right)}\sum _{u=0}^i\sum _{v=0}^j\sum _{y=0}^k{x}_{u,v,y}\to \infty ,\\ \mathit{as}n,m,g\to \infty .\end{array}$$

From last relation follows that $x=\left(x_{n,m,g}\right)$ is not $N_{p,q}^{n,m,g}{C}_{n,m,g}^{\left(\mathrm{1,1,1}\right)}$ summable to $0$.

Theorem 1.2 Let $x=\left(x_{n,m,g}\right)$ be a sequence statistically convergent to $L$ and $\mid x_{n,m,g}-L\mid \le M$ for every $\left(n,m,g\right)\in \mathrm{\mathbb{N}}\times \mathrm{\mathbb{N}}\times \mathrm{\mathbb{N}}$. Then, it converges $N_{p,q}^{n,m,g}{C}_{n,m,g}^{\left(\mathrm{1,1,1}\right)}$-statistically to $L$.

Proof: From the fact that $\left(x_{n,m,g}\right)$ converges statistically to $L$, we have

$$\underset{n,m,g\to \infty }{lim}\frac{\mid i,j,k\le n,m,g:\mid x_{i,j,k}-L\mid \ge \epsilon \}\mid }{\mathit{nmg}}=0.$$

We will denote $B_{\epsilon }=\left\{i,j,k\le n,m,g:|x_{i,j,k}-L|\ge \epsilon \right\}$ and ${\overline{B}}_{\epsilon }=\left\{i,j,k\le n,m,g:|x_{i,j,k}-L|\le \epsilon \right\}$. Then,

$$\begin{array}{l}\mid \frac{1}{R_{n,m,g}}\sum _{i=0}^n\sum _{j=0}^m\sum _{k=0}^g{p}_{i,j,k}{q}_{n-i,m-j,g-k}\frac{1}{\left(i+1\right)\left(j+1\right)\left(k+1\right)}\sum _{u=0}^i\sum _{v=0}^j\sum _{y=0}^k{x}_{u,v,y}-L\mid \\ =\mid \frac{1}{R_{n,m,g}}\sum _{i=0}^n\sum _{j=0}^m\sum _{k=0}^g{p}_{i,j,k}{q}_{n-i,m-j,g-k}\frac{1}{\left(i+1\right)\left(j+1\right)\left(k+1\right)}\sum _{u=0}^i\sum _{v=0}^j\sum _{y=0}^k\left(x_{u,v,y}-L\right)\mid \end{array}$$

$$\begin{array}{l}\le \frac{1}{R_{n,m,g}}{\underset{¯}{\sum _{i=0}^n\sum _{j=0}^m\sum _{k=0}^g}}_{\left(i,j,k\right)\in B_{\epsilon }}{p}_{i,j,k}{q}_{n-i,m-j,g-k}\frac{1}{\left(i+1\right)\left(j+1\right)\left(k+1\right)}\sum _{u=0}^i\sum _{v=0}^j\sum _{y=0}^k\mid x_{u,v,y}-L\mid \\ +\frac{1}{R_{n,m,g}}{\underset{¯}{\sum _{i=0}^n\sum _{j=0}^m\sum _{k=0}^g}}_{\left(i,j,k\right)\in {\overline{B}}_{\epsilon }}{p}_{i,j,k}{q}_{n-i,m-j,g-k}\frac{1}{\left(i+1\right)\left(j+1\right)\left(k+1\right)}\sum _{u=0}^i\sum _{v=0}^j\sum _{y=0}^k\mid x_{u,v,y}-L\mid \\ \le M\frac{1}{R_{n,m,g}}{\underset{¯}{\sum _{i=0}^n\sum _{j=0}^m\sum _{k=0}^g}}_{\left(i,j,k\right)\in B_{\epsilon }}1+\epsilon \le M\frac{C_2}{\mathit{nmg}}{\underset{¯}{\sum _{i=0}^n\sum _{j=0}^m\sum _{k=0}^g}}_{\left(i,j,k\right)\in B_{\epsilon }}1+\epsilon \to 0+\epsilon ,\\ \mathit{as}n,m,g\to \infty ,\end{array}$$

for some constant $C_2$.

Converse of Theorem 1.2 is not true as can be seen in the following example.

Consider that $p_{n,m,g}=\left(n+1\right)\left(m+1\right)\left(g+1\right)$, $\left(q_{n,m,g}\right)=1$ for some $\left(n,m,g\right)\in \mathrm{\mathbb{N}}\cup \left\{0\right\}\times \mathrm{\mathbb{N}}\cup \left\{0\right\}\times \mathrm{\mathbb{N}}\cup \left\{0\right\}$ and define the sequence $x=\left(x_{n,m,g}\right)$ as follows:

$$x_{i,j,k}=\{\begin{array}{l}1,\mathit{\text{for}}i=p^2-p,\dots ,p^2-\mathrm{}j=t^2-t,\dots ,t^2-1\mathit{\text{and}}k=o^2-o,\dots ,o^2-1;\\ -\frac{1}{\mathit{pto}},\mathit{\text{for}}i=p^2,p=2,..j=t^2,t=2,..\mathit{\text{and}}k=o^2,o=2,\dots \\ 0,\mathit{\text{otherwise}}\end{array}$$

Under this conditions, after some basic calculations we get that $x=\left(x_{n,m,g}\right)$ is $N_{p,q}^{n,m,g}{C}_{n,m,g}^{\left(\mathrm{1,1,1}\right)}$-summable to $1$. Therefore, by Theorem 1.2, $x=\left(x_{n,m,g}\right)$ is $N_{p,q}^{n,m,g}{C}_{n,m,g}^{\left(\mathrm{1,1,1}\right)}$-statistically convergent. On the other hand, the sequences $p^2;p=2,3,\dots$, $t^2;t=2,3,\dots$ and $o^2;o=2,3,\dots$ have natural density zero and it is clear that $\mathit{st}$-$\underset{n,m,g}{liminf}{x}_{n,m,g}=0$ and $\mathit{st}$-$\underset{n,m,g}{limsup}{x}_{n,m,g}=1$. Hence, $\left(x_{i,j,k}\right)$ is not statistically convergent.

2. Tauberian theorems under $N_{p,q}^{n,m,g}{C}_{n,m,g}^{\left(\mathrm{1,1,1}\right)}$-statistically convergence

In this section, we show the results that we obtained. Throughout this paper, $R_{{\lambda }_{n,m,g}}$ and $R_{{\lambda }_n,{\lambda }_m,{\lambda }_g}$ will have the same meaning.

Consider that $\mathit{st}$-$limi,j,{\mathit{kx}}_{i,j,k}=L$; $\left(x_{n,m,g}\right)$ is $N_{p,q}^{n,m,g}{C}_{n,m,g}^{\left(\mathrm{1,1,1}\right)}$-statistically convergent and (13) satisfies, then for every $t>1$, is valid the following relation

$$\begin{array}{l}\mathit{st}-\underset{i,j,k}{lim}\frac{1}{R_{{\lambda }_{i,j,k}}-R_{i,j,k}}\sum _{w=i+1}^{{\lambda }_i}\sum _{e=j+1}^{{\lambda }_j}\sum _{r=k+1}^{{\lambda }_k}{p}_{w,e,r}{q}_{{\lambda }_i-w,\mathit{\lambda j}-e,\mathit{\lambda k}-r}\\ \frac{1}{\left(w+1\right)\left(e+1\right)\left(r+1\right)}\sum _{u=0}^i\sum _{v=0}^j\sum _{y=0}^k\left(x_{w,e,r}-x_{i,j,k}\right)=0\end{array}$$E7

and in case where $0<t<1$,

$$\begin{array}{l}\mathit{st}-\underset{i,j,k}{lim}\frac{1}{R_{i,j,k}-R_{{\lambda }_{i,j,k}}}\sum _{w={\lambda }_i+1}^i\sum _{e={\lambda }_j+1}^j\sum _{r={\lambda }_k+1}^k{p}_{w,e,r}{q}_{i-w,j-e,k-r}\\ \frac{1}{\left(w+1\right)\left(e+1\right)\left(r+1\right)}\sum _{u=0}^i\sum _{v=0}^j\sum _{y=0}^k\left(x_{i,j,k}-x_{w,e,r}\right)=0.\end{array}$$E8

The condition given by relation (13) is equivalent to the condition

$$\mathit{st}-\underset{n,m,g\to \infty }{lim}\frac{R_{n,m,g}}{R_{{\lambda }_{n,m,g}}}>1,0<\lambda <1.$$E9

Proof: Suppose that relation (13) is valid, $0<\lambda <1$, $w={\lambda }_n=\left[\mathit{\lambda n}\right]$, $e={\lambda }_m=\left[\mathit{\lambda m}\right]$ and $r={\lambda }_g=\left[\mathit{\lambda g}\right]$, $\left(n,m,g\right)\in \mathrm{\mathbb{N}}\times \mathrm{\mathbb{N}}\times \mathrm{\mathbb{N}}$. Then, it follows that

$$\frac{1}{\lambda }>1\Rightarrow \frac{w}{\lambda }=\frac{\left[\mathit{\lambda n}\right]}{t}\le n,\frac{1}{\lambda }>1\Rightarrow \frac{e}{\lambda }=\frac{\left[\mathit{\lambda m}\right]}{t}\le m\mathit{\text{and}}\frac{1}{\lambda }>1\Rightarrow \frac{r}{\lambda }=\frac{\left[\mathit{\lambda g}\right]}{t}\le g.$$

From above relation and definition of sequences $\left(p_{n,m,g}\right)$ and $\left(q_{n,m,g}\right)$, we have

$$\frac{R_{n,m,g}}{R_{{\lambda }_{n,m,g}}}\ge \frac{R_{\left[\frac{n}{\lambda }\right],\left[\frac{m}{\lambda }\right],\left[\frac{g}{\lambda }\right]}}{R_{{\lambda }_{n,m,g}}}\Rightarrow \mathit{st}-\underset{n,m,g\to \infty }{liminf}\frac{R_{n,m,g}}{R_{{\lambda }_{n,m,g}}}\ge \mathit{st}-\underset{n,m,g\to \infty }{liminf}\frac{R_{\left[\frac{n}{\lambda }\right],\left[\frac{m}{\lambda }\right],\left[\frac{g}{\lambda }\right]}}{R_{{\lambda }_{n,m,g}}}>1.$$

Conversely, suppose that (9) is valid. Now, let $\lambda >1$ be given and let ${\lambda }_1,{\lambda }_2,{\lambda }_3$ be chosen such that $1<{\lambda }_1,{\lambda }_2,{\lambda }_3<\lambda$. Set $w={\lambda }_n=\left[\mathit{\lambda n}\right]$, $e={\lambda }_m=\left[\mathit{\lambda m}\right]$ and $r={\lambda }_g=\left[\mathit{\lambda g}\right]$. From $0<\frac{1}{\lambda }<\frac{1}{{\lambda }_1},\frac{1}{{\lambda }_2},\frac{1}{{\lambda }_3}<1$, it follows that

$$n\le \frac{\mathit{\lambda n}-1}{{\lambda }_1}<\frac{\left[\mathit{\lambda n}\right]}{{\lambda }_1}=\frac{w}{{\lambda }_1},m\le \frac{\mathit{\lambda m}-1}{{\lambda }_2}<\frac{\left[\mathit{\lambda m}\right]}{{\lambda }_2}=\frac{e}{{\lambda }_2}\mathit{\text{and}}g\le \frac{\mathit{\lambda g}-1}{{\lambda }_3}<\frac{\left[\mathit{\lambda g}\right]}{{\lambda }_3}=\frac{r}{{\lambda }_3}$$

provided ${\lambda }_1,{\lambda }_2,{\lambda }_3\le \lambda -\frac{1}{n},\lambda -\frac{1}{m},\lambda -\frac{1}{g}$, which is a case where if $n$, $m$ and $g$ are large enough. Under this condition, we obtain

$$\frac{R_{{\lambda }_{n,m,g}}}{R_{n,m,g}}\ge \frac{R_{{\lambda }_{n,m,g}}}{R_{\left[\frac{w}{{\lambda }_1}\right],\left[\frac{e}{{\lambda }_2}\right],\left[\frac{r}{{\lambda }_3}\right]}}\Rightarrow \mathit{st}-\underset{n,m,g\to \infty }{liminf}\frac{R_{{\lambda }_{n,m,g}}}{R_{n,m,g}}\ge \mathit{st}-\underset{n,m,g\to \infty }{liminf}\frac{R_{{\lambda }_{n,m,g}}}{R_{\left[\frac{w}{{\lambda }_1}\right],\left[\frac{e}{{\lambda }_2}\right],\left[\frac{r}{{\lambda }_3}\right]}}>1.$$

Consider that (13) is satisfied and let $x=\left(x_{i,j,k}\right)$ be a sequence of complex numbers which is $N_{p,q}^{n,m,g}{C}_{n,m,g}^{\left(\mathrm{1,1,1}\right)}$-statistically convergent to $L$. Then,

$$\begin{array}{l}\mathit{st}-\underset{n,m,g}{lim}\frac{1}{R_{{\lambda }_{n,m,g}}-R_{n,m,g}}\sum _{i=n+1}^{{\lambda }_n}\sum _{j=m+1}^{{\lambda }_m}\sum _{k=g+1}^{{\lambda }_g}{p}_{i,j,k}{q}_{{\lambda }_n-i,{\lambda }_m-j,{\lambda }_g-k}\\ \frac{1}{\left(i+1\right)\left(j+1\right)\left(k+1\right)}\sum _{u=0}^i\sum _{v=0}^j\sum _{y=0}^k{x}_{u,v,y}=L\mathit{\text{for}}\lambda >1\end{array}$$E10

and

$$\begin{array}{l}\mathit{st}-\underset{n,m,g}{lim}\frac{1}{R_{n,m,g}-R_{{\lambda }_{n,m,g}}}\sum _{i={\lambda }_n+1}^n\sum _{j={\lambda }_m+1}^m\sum _{k={\lambda }_g+1}^g{p}_{i,j,k}{q}_{n-i,m-j,g-k}\\ \frac{1}{\left(i+1\right)\left(j+1\right)\left(k+1\right)}\sum _{u=0}^i\sum _{v=0}^j\sum _{y=0}^k{x}_{u,v,y}=L\mathit{\text{for}}0<\lambda <1.\end{array}$$E11

Proof: We begin proving the case (10), i.e. when $\lambda >1$. Then, we have

$$\begin{array}{l}\frac{1}{R_{{\lambda }_{n,m,g}}-R_{n,m,g}}\sum _{i=n+1}^{{\lambda }_n}\sum _{j=m+1}^{{\lambda }_m}\sum _{k=g+1}^{{\lambda }_g}{p}_{i,j,k}{q}_{{\lambda }_n-i,{\lambda }_m-j,{\lambda }_g-k}\frac{1}{\left(i+1\right)\left(j+1\right)\left(k+1\right)}\sum _{u=0}^i\sum _{v=0}^j\sum _{y=0}^k\left(x_{u,v,y}-L\right)\\ =\begin{array}{l}\frac{R_{{\lambda }_{n,m,g}}}{R_{{\lambda }_{n,m,g}}-R_{n,m,g}}\frac{1}{R_{{\lambda }_{n,m,g}}}\sum _{i=n+1}^{{\lambda }_n}\sum _{j=m+1}^{{\lambda }_m}\sum _{k=g+1}^{{\lambda }_g}{p}_{i,j,k}{q}_{{\lambda }_n-i,{\lambda }_m-j,{\lambda }_g-k}\frac{1}{\left(i+1\right)\left(j+1\right)\left(k+1\right)}\\ \sum _{u=0}^i\sum _{v=0}^j\sum _{y=0}^k\left(x_{u,v,y}-L\right)-\frac{R_{n,m,g}}{R_{{\lambda }_{n,m,g}}-R_{n,m,g}}\frac{1}{R_{n,m,g}}\sum _{i=0}^n\sum _{j=0}^m\sum _{k=0}^g{p}_{i,j,k}{q}_{{\lambda }_n-i,{\lambda }_m-j,{\lambda }_g-k}\\ \frac{1}{\left(i+1\right)\left(j+1\right)\left(k+1\right)}\sum _{u=0}^i\sum _{v=0}^j\sum _{y=0}^k\left(x_{u,v,y}-L\right)\end{array}\\ =\begin{array}{l}\frac{R_{{\lambda }_{n,m,g}}}{R_{{\lambda }_{n,m,g}}-R_{n,m,g}}\frac{1}{R_{{\lambda }_{n,m,g}}}\sum _{i=n+1}^{{\lambda }_n}\sum _{j=m+1}^{{\lambda }_m}\sum _{k=g+1}^{{\lambda }_g}{p}_{i,j,k}{q}_{{\lambda }_n-i,{\lambda }_m-j,{\lambda }_g-j}\\ \frac{1}{\left(i+1\right)\left(j+1\right)\left(k+1\right)}\sum _{u=0}^i\sum _{v=0}^j\sum _{y=0}^k\left(x_{u,v,y}-L\right)-\frac{R_{n,m,g}}{R_{{\lambda }_{n,m,g}}-R_{n,m,g}}\frac{1}{R_{n,m,g}}\sum _{i=0}^n\sum _{j=0}^m\sum _{k=0}^g{p}_{i,j,k}\\ \left(q_{{\lambda }_n-i,{\lambda }_m-j,{\lambda }_g-k}+q_{n-i,m-j,g-k}-q_{n-i,m-j,g-k}\right)\frac{1}{\left(i+1\right)\left(j+1\right)\left(k+1\right)}\sum _{u=0}^i\sum _{v=0}^j\sum _{y=0}^k\left(x_{u,v,y}-L\right)\end{array}\\ =\begin{array}{l}\frac{R_{{\lambda }_{n,m,g}}}{R_{{\lambda }_{n,m,g}}-R_{n,m,g}}\frac{1}{R_{{\lambda }_{n,m,g}}}\sum _{i=n+1}^{{\lambda }_n}\sum _{j=m+1}^{{\lambda }_m}\sum _{k=g+1}^{{\lambda }_g}{p}_{i,j,k}{q}_{{\lambda }_n-i,{\lambda }_m-j,{\lambda }_g-k}\\ \frac{1}{\left(i+1\right)\left(j+1\right)\left(k+1\right)}\sum _{u=0}^i\sum _{v=0}^j\sum _{y=0}^k\left(x_{u,v,y}-L\right)-\frac{R_{n,m,g}}{R_{{\lambda }_{n,m,g}}-R_{n,m,g}}\frac{1}{R_{n,m,g}}\sum _{i=0}^n\sum _{j=0}^m\sum _{k=0}^g{p}_{i,j,k}\\ q_{n-i,m-j,g-k}\frac{1}{\left(i+1\right)\left(j+1\right)\left(k+1\right)}\sum _{u=0}^i\sum _{v=0}^j\sum _{y=0}^k{x}_{u,v,y}-\frac{R_{n,m,g}}{R_{{\lambda }_{n,m,g}}-R_{n,m,g}}\frac{1}{R_{n,m,g}}\sum _{i=0}^n\sum _{j=0}^m\sum _{k=0}^g{p}_{i,j,k}\\ \left(q_{{\lambda }_n-i,{\lambda }_m-j,{\lambda }_g-k}-q_{n-i,m-j,g-k}\right)\frac{1}{\left(i+1\right)\left(j+1\right)\left(k+1\right)}\sum _{u=0}^i\sum _{v=0}^j\sum _{y=0}^k\left(x_{u,v,y}-L\right).\end{array}\end{array}$$E12

From (12), definition of the sequence $\left(q_{n,m,g}\right)$ and relation $\underset{n,m,g}{limsup}\frac{R_{{\lambda }_{n,m,g}}}{R_{{\lambda }_{n,m,g}}-R_{n,m,g}}<\infty$, we get (10).

Prove of (11) is made similarly to the prove of (10).

In the following theorem, we characterize the converse implication when the statistically convergence follows from its $N_{p,q}^{n,m,g}{C}_{n,m,g}^{\left(\mathrm{1,1,1}\right)}$− statistically convergence.

Theorem 1.3 Let $\left(p_{n,m,g}\right)$ and $\left(q_{n,m,g}\right)$ be two non-negative real sequences and

$$\mathit{st}-\underset{n,m,g\to \infty }{liminf}\frac{R_{{\lambda }_{n,m,g}}}{R_{n,m,g}}>1\mathit{\text{for}}\mathit{\text{every}}\lambda >1,$$E13

where ${\lambda }_{n,m,g}={\lambda }_n{\lambda }_m{\lambda }_g=\left[\mathit{\lambda n}\right]\left[\mathit{\lambda m}\right]\left[\mathit{\lambda g}\right]$ denotes the integral part of $\mathit{\text{λnλmλg}}$ for every $\left(n,m,g\right)\in \mathrm{\mathbb{N}}\times \mathrm{\mathbb{N}}\times \mathrm{\mathbb{N}}$, and let $\left(x_{n,m,g}\right)$ be a sequence of real numbers which is $N_{p,q}^{n,m,g}{C}_{n,m,g}^{\left(\mathrm{1,1,1}\right)}$-statistically convergent to a finite number $L$. Then, $\left(x_{n,m,g}\right)$ is $\mathit{st}$-convergent to the same number $L$ if and only if the following two conditions hold

$$\begin{array}{l}\underset{\lambda >1}{inf}\underset{n,m,g}{limsup}\frac{1}{R_{n,m,g}}\mid \{i,j,k\le R_{n,m,g}:\frac{1}{R_{{\lambda }_{i,j,k}}-R_{i,j,k}}\sum _{w=i+1}^{{\lambda }_i}\sum _{e=j+1}^{{\lambda }_j}\sum _{r=k+1}^{{\lambda }_k}{p}_{w,e,r}{q}_{{\lambda }_i-w,{\lambda }_j-e,{\lambda }_k-r}\\ \frac{1}{\left(w+1\right)\left(e+1\right)\left(r+1\right)}\sum _{u=0}^i\sum _{v=0}^j\sum _{y=0}^k\left(x_{w,e,r}-x_{i,j,k}\right)\le -\epsilon \}\mid =0,\end{array}$$E14

and

$$\begin{array}{l}\underset{0<\lambda <1}{inf}\underset{n,m,g}{limsup}\frac{1}{R_{n,m,g}}\mid \{i,j,k\le R_{n,m,g}:\frac{1}{R_{i,j,k}-R_{{\lambda }_{i,j,k}}}\sum _{w={\lambda }_i+1}^i\sum _{e={\lambda }_j+1}^j\sum _{r={\lambda }_k+1}^k{p}_{w,e,r}{q}_{i-w,j-e,k-r}\\ \frac{1}{\left(w+1\right)\left(e+1\right)\left(r+1\right)}\sum _{u=0}^i\sum _{v=0}^j\sum _{y=0}^k\left(x_{i,j,k}-x_{w,e,r}\right)\le -\epsilon \}\mid =0.\end{array}$$E15

Proof: Necessity: Suppose that ${lim}_{n,m,g\to \infty }{x}_{n,m,g}=L$ and (13) holds. By Proposition 2, we have

$$\begin{array}{l}\underset{n,m,g\to \infty }{lim}\frac{1}{R_{{\lambda }_{n,m,g}}-R_{n,m,g}}\sum _{i=n+1}^{{\lambda }_n}\sum _{j=m+1}^{{\lambda }_m}\sum _{k=g+1}^{{\lambda }_g}{p}_{i,j,k}{q}_{{\lambda }_n-i,{\lambda }_m-j,{\lambda }_g-k}\\ \frac{1}{\left(i+1\right)\left(j+1\right)\left(k+1\right)}\sum _{u=0}^i\sum _{v=0}^j\sum _{y=0}^k\left(x_{u,v,y}-x_{n,m,g}\right)\\ =\underset{n,m,g\to \infty }{lim}\left\{\right(\frac{1}{R_{{\lambda }_{n,m,g}}-R_{n,m,g}}\sum _{i=n+1}^{{\lambda }_n}\sum _{j=m+1}^{{\lambda }_m}\sum _{k=g+1}^{{\lambda }_g}{p}_{i,j,k}{q}_{{\lambda }_n-i,{\lambda }_m-j,{\lambda }_g-k}\\ \frac{1}{\left(i+1\right)\left(j+1\right)\left(k+1\right)}\sum _{u=0}^i\sum _{v=0}^j\sum _{y=0}^k{x}_{u,v,y})-x_{n,m,g}\}=0,\end{array}$$

for every $\lambda >1$. In case where $0<\lambda <1$, we have that

$$\begin{array}{l}\underset{n,m,g\to \infty }{lim}\frac{1}{R_{n,m,g}-R_{{\lambda }_{n,m,g}}}\sum _{i={\lambda }_n+1}^n\sum _{j={\lambda }_m+1}^m\sum _{k={\lambda }_g+1}^g{p}_{i,j,k}{q}_{n-i,m-j,g-k}\\ \frac{1}{\left(i+1\right)\left(j+1\right)\left(k+1\right)}\sum _{u=0}^i\sum _{v=0}^j\sum _{y=0}^k\left(x_{n,m,g}-x_{u,v,y}\right)\\ =\underset{n,m,g\to \infty }{lim}\{x_{n,m,g}-(\frac{1}{R_{n,m,g}-R_{{\lambda }_{n,m,g}}}\sum _{i={\lambda }_n+1}^n\sum _{j={\lambda }_m+1}^m\sum _{k={\lambda }_g+1}^g{p}_{i,j,k}{q}_{n-i,m-j,g-k}\\ \frac{1}{\left(i+1\right)\left(j+1\right)\left(k+1\right)}\sum _{u=0}^i\sum _{v=0}^j\sum _{y=0}^k{x}_{u,v,y}\left)\right\}=0.\end{array}$$

Sufficiency: Consider that (14) and ((15) are satisfied. In what follows, we will prove that $\underset{n,m,g\to \infty }{lim}{x}_{n,m,g}=L$. Given any $\epsilon >0$, by (14) we can choose ${\lambda }_1>0$ such that

$$\begin{array}{l}\underset{n,m,g\to \infty }{liminf}\frac{1}{R_{{\lambda }_{n_1},{\lambda }_{m_1},{\lambda }_{g_1}-R_{n,m,g}}}\sum _{i=n+1}^{{\lambda }_{n_1}}\sum _{j=m+1}^{{\lambda }_{m_1}}\sum _{k=g+1}^{{\lambda }_{g_1}}{p}_{i,j,k}{q}_{{\lambda }_n-i,{\lambda }_m-j,{\lambda }_g-k}\\ \frac{1}{\left(i+1\right)\left(j+1\right)\left(k+1\right)}\sum _{u=0}^i\sum _{v=0}^j\sum _{y=0}^k\left(x_{u,v,y}-x_{n,m,g}\right)\ge -\epsilon ,\end{array}$$E16

where ${\lambda }_{n_1}=\left[\lambda n_1\right]$, ${\lambda }_{m_1}=\left[\lambda m_1\right]$ and ${\lambda }_{g_1}=\left[\lambda g_1\right]$. By the assumed summability $N_{p,q}^{n,m,g}{C}_{n,m,g}^{\left(\mathrm{1,1,1}\right)}$ of $\left(x_{n,m,g}\right)$, Proposition 2 and (16), we have

$$\underset{n,m,g\to \infty }{limsup}{x}_{n,m,g}\le L+\epsilon ,$$E17

for any $\lambda >1$.

On the other hand, if $0<\lambda <1$, for every $\epsilon >0$, we can choose $0<{\lambda }_2<1$ such that

$$\begin{array}{l}m\underset{n,m,g\to \infty }{liminf}\frac{1}{R_{n,m,g}-R_{{\lambda }_{n_2},{\lambda }_{m_2},{\lambda }_{g_2}}}\sum _{i={\lambda }_{n_2}+1}^n\sum _{j={\lambda }_{m_2}+1}^m\sum _{k={\lambda }_{g_2}+1}^g{p}_{i,j,k}{q}_{n-i,m-j,g-k}\\ \frac{1}{\left(i+1\right)\left(j+1\right)\left(k+1\right)}\sum _{u=0}^i\sum _{v=0}^j\sum _{y=0}^k\left(x_{n,m,g}-x_{u,v,y}\right)\ge -\epsilon ,\end{array}$$E18

where ${\lambda }_{n_2}=\left[\lambda n_2\right]$, ${\lambda }_{m_2}=\left[\lambda m_2\right]$ and ${\lambda }_{g_2}=\left[\lambda g_2\right]$. By the assumed summability $N_{p,q}^{n,m,g}{C}_{n,m,g}^{\left(\mathrm{1,1,1}\right)}$ of $\left(x_{n,m,g}\right)$, Proposition 2 and (18), we have

$$\underset{n,m,g\to \infty }{liminf}{x}_{n,m,g}\ge L-\epsilon ,$$E19

for any $0<\lambda <1$.

Since $\epsilon >0$ is arbitrary, combining (17) and (19), we obtain

$$\underset{n,m,g\to \infty }{lim}{x}_{n,m,g}=L.$$

In the following theorem, we will consider the case where $x=\left(x_{n,m,g}\right)$ is a sequence of complex numbers.

Theorem 1.4 Let (13) be satisfied and let $\left(x_{n,m,g}\right)$ be a sequence of complex numbers which is $N_{p,q}^{n,m,g}{C}_{n,m,g}^{\left(\mathrm{1,1,1}\right)}$-statistically convergent to a finite number $L$. Then, $\left(x_{n,m,g}\right)$ is convergent to the same number $L$ if and only if the following two conditions hold

$$\begin{array}{l}\underset{\lambda >1}{inf}\underset{n,m,g}{limsup}\frac{1}{R_{n,m,g}}\mid \{i,j,k\le R_{n,m,g}:\frac{1}{R_{{\lambda }_{i,j,k}}-R_{i,j,k}}\sum _{w=i+1}^{{\lambda }_i}\sum _{e=j+1}^{{\lambda }_j}\sum _{r=k+1}^{{\lambda }_k}{p}_{w,e,r}{q}_{{\lambda }_i-w,{\lambda }_j-e,{\lambda }_k-r}\\ \frac{1}{\left(w+1\right)\left(e+1\right)\left(k+1\right)}\sum _{u=0}^i\sum _{v=0}^j\sum _{y=0}^k\left(x_{w,e,r}-x_{i,j,k}\right)\ge \epsilon \}\mid =0,\end{array}$$E20

and

$$\begin{array}{l}\underset{0<\lambda <1}{inf}\underset{n,m,g}{limsup}\frac{1}{R_{n,m,g}}\mid \{i,j,k\le R_{n,m,g}:\frac{1}{R_{i,j,k}-R_{{\lambda }_{i,j,k}}}\sum _{w={\lambda }_i+1}^i\sum _{e={\lambda }_j+1}^j\sum _{r={\lambda }_k+1}^k{p}_{w,e,r}{q}_{i-w,j-e,k-r}\\ \frac{1}{\left(w+1\right)\left(e+1\right)\left(r+1\right)}\sum _{u=0}^i\sum _{v=0}^j\sum _{y=0}^k\left(x_{i,j,k}-x_{w,e,r}\right)\ge \epsilon \}\mid =0.\end{array}$$E21

Proof: Necessity: If both (2) and (6) hold, then Proposition 2 yields (20) for every $\lambda >1$ and (21) for every $0<\lambda <1$.

Sufficiency: Suppose that (2), (13) and one of the conditions (20) and (21) are satisfied. For any given $\epsilon >0$, there exists ${\lambda }_1>0$ such that

$$\begin{array}{l}\underset{n,m,g\to \infty }{limsup}\mid \frac{1}{R_{{\lambda }_{n_1},{\lambda }_{m_1},{\lambda }_{g_1}}-R_{n,m,g}}\sum _{i=n+1}^{{\lambda }_{n_1}}\sum _{j=m+1}^{{\lambda }_{m_1}}\sum _{k=g+1}^{{\lambda }_{g_1}}{p}_{i,j,k}{q}_{{\lambda }_{n_1}-i,{\lambda }_{m_1}-j,{\lambda }_{g_1}-k}\\ \frac{1}{\left(i+1\right)\left(j+1\right)\left(k+1\right)}\sum _{u=0}^i\sum _{v=0}^j\sum _{y=0}^k\left(x_{u,v,y}-x_{n,m,g}\right)\mid \le \epsilon ,\end{array}$$

where ${\lambda }_{n_1}=\left[\lambda n_1\right]$, ${\lambda }_{m_1}=\left[\lambda m_1\right]$ and ${\lambda }_{g_1}=\left[\lambda g_1\right]$. Taking into account fact that $\left(x_{n,m,g}\right)$ is $N_{p,q}^{n,m,g}{C}_{n,m,g}^{\left(\mathrm{1,1,1}\right)}$ summbale to $L$ and Proposition 2, we have the following estimation

$$\begin{array}{l}\underset{n,m,g\to \infty }{limsup}\mid L-x_{n,m,g}\mid \\ \le \underset{n,m,g\to \infty }{limsup}\mid L-\frac{1}{R_{{\lambda }_{n_1},{\lambda }_{m_1},{\lambda }_{g_1}}-R_{n,m}}\sum _{i=n+1}^{{\lambda }_{n_1}}\sum _{j=m+1}^{{\lambda }_{m_1}}\sum _{k=g+1}^{{\lambda }_{g_1}}{p}_{i,j,k}{q}_{{\lambda }_{n_1}-i,{\lambda }_{m_1}-j,{\lambda }_{g_1}-k}\\ \frac{1}{\left(i+1\right)\left(j+1\right)\left(k+1\right)}\sum _{u=0}^i\sum _{v=0}^j\sum _{y=0}^k{x}_{u,v,y}\mid \\ +\underset{n,m,g\to \infty }{limsup}\mid L-\frac{1}{R_{{\lambda }_{n_1},{\lambda }_{m_1},{\lambda }_{g_1}}-R_{n,m}}\sum _{i=n+1}^{{\lambda }_{n_1}}\sum _{j=m+1}^{{\lambda }_{m_1}}\sum _{k=g+1}^{{\lambda }_{g_1}}{p}_{i,j,k}{q}_{{\lambda }_{n_1}-i,{\lambda }_{m_1}-j,{\lambda }_{g_1}-k}\\ \frac{1}{\left(i+1\right)\left(j+1\right)\left(k+1\right)}\sum _{u=0}^i\sum _{v=0}^j\sum _{y=0}^k\left(x_{u,v,y}-x_{n,m,g}\right)\mid \\ \le \epsilon .\end{array}$$

For a given $\epsilon >0$, there exists ${\lambda }_2>0$ such that

$$\begin{array}{l}\underset{n,m,g\to \infty }{limsup}\mid \frac{1}{R_{n,m,g}-R_{{\lambda }_{n_2},{\lambda }_{m_2},{\lambda }_{g_2}}}\sum _{i={\lambda }_{n_2}+1}^n\sum _{j={\lambda }_{m_2}+1}^m\sum _{k={\lambda }_{g_2}+1}^g{p}_{i,j,k}{q}_{n-i,m-j,g-k}\\ \frac{1}{\left(i+1\right)\left(j+1\right)\left(k+1\right)}\sum _{u=0}^i\sum _{v=0}^j\sum _{y=0}^k\left(x_{n,m,g}-x_{u,v,y}\right)\mid \le \epsilon ,\end{array}$$

where ${\lambda }_{n_2}=\left[\lambda n_2\right]$, ${\lambda }_{m_2}=\left[\lambda m_2\right]$ and ${\lambda }_{g_2}=\left[\lambda g_2\right]$. Taking into account fact that $\left(x_{n,m,g}\right)$ is $N_{p,q}^{n,m,g}{C}_{n,m,g}^{\left(\mathrm{1,1,1}\right)}$ summbale to $L$ and Proposition 2, we obtain the following

$$\begin{array}{l}\underset{n,m,g\to \infty }{limsup}\mid L-x_{n,m,g}\mid \\ \underset{n,m\to \infty }{limsup}\mid L-\frac{1}{R_{n,m,g}-R_{{\lambda }_{n_2},{\lambda }_{m_2},{\lambda }_{g_2}}}\sum _{i={\lambda }_{n_2}+1}^n\sum _{j={\lambda }_{m_2}+1}^m\sum _{k={\lambda }_{g_2}+1}^g{p}_{i,j,k}{q}_{n-i,m-j,g-k}\\ \frac{1}{\left(i+1\right)\left(j+1\right)\left(k+1\right)}\sum _{u=0}^i\sum _{v=0}^j\sum _{y=0}^k{x}_{u,v,y}\mid \\ +\underset{n,m,g\to \infty }{limsup}\mid \frac{1}{R_{n,m,g}-R_{{\lambda }_{n_2},{\lambda }_{m_2},{\lambda }_{g_2}}}\sum _{i={\lambda }_{n_2}+1}^n\sum _{j={\lambda }_{m_2}+1}^m\sum _{k={\lambda }_{g_2}+1}^g{p}_{i,j,k}{q}_{n-i,m-j,g-k}\\ \frac{1}{\left(i+1\right)\left(j+1\right)\left(k+1\right)}\sum _{u=0}^i\sum _{v=0}^j\sum _{y=0}^k\left(x_{n,m,g}-x_{u,v,y}\right)\mid \\ \le \epsilon .\end{array}$$

Since $\epsilon >0$ in either case, we get

$$\underset{n,m,g\to \infty }{lim}{x}_{n,m,g}=L.$$

3. Conclusion

In this paper, we have defined and proved new Tauberian theorems under triple statistically Nörlund-Cesáro summability, as a consequence of results showed in 2, some theorems, lemmas and corollaries can be defined and proved similarly by using $\left(\mathrm{1,0,0}\right)$, $\left(\mathrm{0,1,0}\right)$_, and $\left(\mathrm{0,0,1}\right)$ method of summability. It is well know that Tauberian theorems for single sequences of single variable have been achieved a high degree of development; however, it is still in its infancy for triple sequences. For that reason, the results established in this paper can be extended and studied in some inclusion, Tauberian type theorems and Tauberian convexity type for certain families of generalized Nörlund.