On some spaces of almost lacunary convergent sequences derived by Riesz mean and weighted almost lacunary statistical convergence in a real n-normed space

Abstract

In this paper, we introduce some new spaces of almost convergent sequences derived by Riesz mean and the lacunary sequence in a real n-normed space. By combining the definitions of lacunary sequence and Riesz mean, we obtain a new concept of statistical convergence which will be called weighted almost lacunary statistical convergence in a real n-normed space. We examine some connections between this notion with the concept of almost lacunary statistical convergence and weighted almost statistical convergence, where the base space is a real n-normed space.

MSC:40C05, 40A35, 46A45, 40A05, 40F05.

1 Introduction

The concept of 2-normed space has been initially introduced by Gähler [1]. Later, this concept was generalized to the concept of n-normed spaces by Misiak [2]. Since then, many others have studied these concepts and obtained various results [3–10].

The idea of statistical convergence was given by Zygmund [11] in 1935, in order to extend the convergence of sequences. The concept was formally introduced by Fast [12] and Steinhaus [13] and later on by Schoenberg [14], and also independently by Buck [15]. Many years later, it has been discussed in the theory of Fourier analysis, ergodic theory, and number theory under different names. In 1993, Fridy and Orhan [16] introduced the concept of lacunary statistical convergence. Statistical convergence has been generalized to the concept of a 2-normed space by Gürdal and Pehlivan [3] and to the concept of an n-normed space by Reddy [9].

Moricz and Orhan [17] have defined the concept of statistical summability $(R,p_r)$. Later on, Karakaya and Chishti [18] have used $(R,p_r)$-summability to generalize the concept of statistical convergence and have called this new method weighted statistical convergence. Mursaleen et al. [19] have altered the definition of weighted statistical convergence and have found its relation with the concept of statistical $(R,p_r)$-summability. In general, the statistical convergence of weighted mean is studied as a regular matrix transformation. In [18] and [19], the concept of statistical convergence is generalized by using a Riesz summability method and it is called weighted statistical convergence. For more details related to this topic, we may refer to [5, 20–23].

In this paper, we introduce some new spaces of almost convergent sequences derived by Riesz mean and lacunary sequence in a real n-normed space. By combining the definitions of lacunary sequence and Riesz mean, we obtain a new concept of statistical convergence, which will be called weighted almost lacunary statistical convergence in a real n-normed space. We examine some connections between this notion with the concept of almost lacunary statistical convergence and weighted almost statistical convergence, where the base space is a real n-normed space.

2 Definitions and preliminaries

Let K be a subset of natural numbers ℕ and we denote the set $K_n=\{j\in K:j\le n\}$. The cardinality of $K_n$ is denoted by $|K_n|$. The natural density of K is given by $\delta (K):={lim}_r\frac{1}{r}|K_r|$, if it exists. The sequence $x=(x_j)$ is statistically convergent to ξ provided that, for every $\epsilon >0$, the set $K=K(\epsilon ):=\{j\in \mathbb{N}:|x_j-\xi |\ge \epsilon \}$ has natural density zero.

Let $(p_k)$ be a sequence of non-negative real numbers and $P_r=p_1+p_2+\cdots +p_r$ for $r\in \mathbb{N}$. Then the Riesz transformation of $x=(x_k)$ is defined as

$$t_r:=\frac{1}{P_r}\sum _{k=1}^r{p}_k{x}_k.$$

(2.1)

If the sequence $t_r$ has a finite limit ξ, then the sequence x is said to be $(R,p_r)$-convergent to ξ. Let us note that if $P_r\to \mathrm{\infty }$ as $r\to \mathrm{\infty }$ then the Riesz transformation is a regular summability method, that is, it transforms every convergent sequence to convergent sequence and preserves the limit.

If $p_k=1$ for all $k\in \mathbb{N}$ in (2.1), then the Riesz mean reduces to the Cesaro mean $C_1$ of order one.

By a lacunary sequence $\theta =(k_r)$, where $k_0=0$, we will mean an increasing sequence of non-negative integers with $k_r-k_{r-1}\to \mathrm{\infty }$ as $r\to \mathrm{\infty }$. The intervals determined by θ will be denoted by $I_r=(k_{r-1},k_r]$. We write $h_r=k_r-k_{r-1}$. The ratio $\frac{k_r}{k_{r-1}}$ will be denoted by $q_r$.

Throughout the paper, we will use the following notations, which have been defined in [24].

Let $\theta =(k_r)$ be a lacunary sequence, $(p_k)$ be a sequence of positive real numbers such that $H_r:={\sum }_{k\in I_r}{p}_k$, $P_{k_r}:={\sum }_{k\in (0,k_r]}{p}_k$, $P_{k_{r-1}}:={\sum }_{k\in (0,k_{r-1}]}{p}_k$, $Q_r:=\frac{P_{k_r}}{P_{k_r-1}}$, $P_0=0$ and the intervals determined by θ and $(p_k)$ are denoted by $I_r^{\prime }=(P_{k_{r-1}},P_{k_r}]$, $H_r=P_{k_r}-P_{k_{r-1}}$. If $p_k=1$ for all $k\in \mathbb{N}$, then $H_r$, $P_{k_r}$, $P_{k_{r-1}}$, $Q_r$ and $I_r^{\prime }$ reduce to $h_r$, $k_r$, $k_{r-1}$, $q_r$ and $I_r$, respectively.

If $\theta =(k_r)$ is a lacunary sequence and $P_r\to \mathrm{\infty }$ as $r\to \mathrm{\infty }$, then ${\theta }^{\prime }=(P_{k_r})$ is a lacunary sequence, that is, $P_0=0$, $0<P_{k_{r-1}}<P_{k_r}$ and $H_r=P_{k_r}-P_{k_{r-1}}\to \mathrm{\infty }$ as $r\to \mathrm{\infty }$.

Throughout the paper, we will take $P_r\to \mathrm{\infty }$ as $r\to \mathrm{\infty }$, unless otherwise stated.

Lorentz [25] has proved that a sequence x is almost convergent to a number ξ if and only if $t_{km}(x)\to \xi$ as $k\to \mathrm{\infty }$, uniformly in m, where

$$t_{km}(x)=\frac{x_m+x_{m+1}+\cdots +x_{m+k-1}}{k},k\in \mathbb{N},m\ge 0.$$

(2.2)

We write $f-limx=\xi$ if x is almost convergent to ξ. Maddox [26] has defined $x=(x_j)$ to be strongly almost convergent to a number ξ if and only if $t_{km}(|x-\xi e|)\to 0$ as $k\to \mathrm{\infty }$, uniformly in m, where $x-\xi e=(x_j-\xi )$ for all j and $e=(1,1,\dots )$.

Let $n\in \mathbb{N}$ and X be a real vector space of dimension $d\ge n\ge 2$. A real-valued function $\parallel \cdot ,\dots ,\cdot \parallel :X^n\to \mathbb{R}$ satisfying the following conditions is called an n-norm on X and the pair $(X,\parallel \cdot ,\dots ,\cdot \parallel )$ is called a linear n-normed space:

• (1) $\parallel x_1,\dots ,x_n\parallel =0$ if and only if $x_1,\dots ,x_n$ are linearly dependent,

• (2) $\parallel x_1,\dots ,x_n\parallel$ is invariant under permutation,

• (3) $\parallel \alpha x_1,\dots ,x_{n-1},x_n\parallel =|\alpha |\parallel x_1,\dots ,x_{n-1},x_n\parallel$ for any $\alpha \in \mathbb{R}$,

• (4) $\parallel x_1,\dots ,x_{n-1},y+z\parallel ⩽\parallel x_1,\dots ,x_{n-1},y\parallel +\parallel x_1,\dots ,x_{n-1},z\parallel$, for all $y,z,x_1,\dots ,x_{n-1}\in X$.

A sequence $x=(x_j)$ in an n-normed space $(X,\parallel \cdot ,\dots ,\cdot \parallel )$ is said to be convergent to some $\xi \in X$ in the n-norm if for each $\epsilon >0$ there exists a positive integer $j_0=j_0(\epsilon )$ such that $\parallel x_j-\xi ,z_1,\dots ,z_{n-1}\parallel <\epsilon$ for all $j\ge j_0$ and for every nonzero $z_1,\dots ,z_{n-1}\in X$.

A sequence $x=(x_j)$ is said to be statistically convergent to ξ if for every $\epsilon >0$ the set $K:=\{j\in \mathbb{N}:\parallel x_j-\xi ,z_1,\dots ,z_{n-1}\parallel \ge \epsilon \}$ has natural density zero for every nonzero $z_1,\dots ,z_{n-1}\in X$, in other words, $x=(x_j)$ is statistically convergent to ξ in n-normed space $(X,\parallel \cdot ,\dots ,\cdot \parallel )$ if ${lim}_{j\to \mathrm{\infty }}\frac{1}{j}|\{j\in \mathbb{N}:\parallel x_j-\xi ,z_1,\dots ,z_{n-1}\parallel \ge \epsilon \}|=0$, for every nonzero $z_1,\dots ,z_{n-1}\in X$. For $\xi =0$, we say this is statistically null.

3 Main results

Throughout the paper $w(X)$, $l_{{}^{\mathrm{\infty }}}(X)$ denote the spaces of all and bounded X valued sequence spaces, respectively, where $(X,\parallel \cdot ,\dots ,\cdot \parallel )$ is a real n-normed space.

The set of all almost convergent sequences and strongly almost convergent sequences with respect to the n-norm $\parallel \cdot ,\cdot \parallel$ are denoted by F and $[F]$, respectively, as follows:

$$F=\left\{\begin{array}{l}x\in l_{\mathrm{\infty }}(X):{lim}_{k\to \mathrm{\infty }}\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel =0,\text{ uniformly in }m,\\ \text{for every nonzero }{z}_1,\dots ,z_{n-1}\in X\end{array}\right\},$$

and

$$[F]=\left\{\begin{array}{l}x\in l_{\mathrm{\infty }}(X):{lim}_{k\to \mathrm{\infty }}{t}_{km}(\parallel x-\xi e,z_1,\dots ,z_{n-1}\parallel )=0,\text{ uniformly in }m,\\ \text{for every nonzero }{z}_1,\dots ,z_{n-1}\in X\end{array}\right\},$$

where $t_{km}(x)$ is defined as in (2.2). We write $F-limx=\xi$ if x is almost convergent to ξ with respect to the n-norm and $[F]-limx=\xi$ if x is strongly almost convergent to ξ with respect to the n-norm. It is easy to see that the inclusions $[F]\subset F\subset l_{{}^{{}_{{}^{\mathrm{\infty }}}}}(X)$ hold.

Now, we define some new sequence spaces in a real n-normed space as follows:

$$\begin{array}{c}{[\tilde{R},p_r,\theta ]}_n=\left\{\begin{array}{l}x:{lim}_{r\to \mathrm{\infty }}\parallel \frac{1}{H_r}{\sum }_{k\in I_r}{p}_k{t}_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel =0,\text{ uniformly in }m,\\ \text{for some }\xi \text{ and for every nonzero }{z}_1,\dots ,z_{n-1}\in X\end{array}\right\},\hfill \\ {(\tilde{R},p_r,\theta )}_n=\left\{\begin{array}{l}x:{lim}_{r\to \mathrm{\infty }}\frac{1}{H_r}{\sum }_{k\in I_r}{p}_k\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel =0,\text{ uniformly in }m,\\ \text{for some }\xi \text{ and for every nonzero }{z}_1,\dots ,z_{n-1}\in X\end{array}\right\},\hfill \\ |\tilde{R},p_r,\theta {|}_n=\left\{\begin{array}{l}x:{lim}_{r\to \mathrm{\infty }}\frac{1}{H_r}{\sum }_{k\in I_r}{p}_k{t}_{km}(\parallel x-\xi e,z_1,\dots ,z_{n-1}\parallel )=0,\text{ uniformly in }m,\\ \text{for some }\xi \text{ and for every nonzero }{z}_1,\dots ,z_{n-1}\in X\end{array}\right\}.\hfill \end{array}$$

The following results are obtained for some special cases:

• (1) If we take $m=0$ then the sequence spaces above are reduced to the sequence spaces ${[C_1,\theta ]}_n$, ${(C_1,\theta )}_n$, $|C_1,\theta {|}_n$, respectively as follows:

  $$\begin{array}{c}{[C_1,\theta ]}_n=\left\{\begin{array}{l}x:{lim}_{r\to \mathrm{\infty }}\parallel \frac{1}{H_r}{\sum }_{k\in I_r}{p}_k{t}_{k0}(x-\xi e),z_1,\dots ,z_{n-1}\parallel =0,\\ \text{for some }\xi \text{ and for every nonzero }{z}_1,\dots ,z_{n-1}\in X\end{array}\right\},\hfill \\ {(C_1,\theta )}_n=\left\{\begin{array}{l}x:{lim}_{r\to \mathrm{\infty }}\frac{1}{H_r}{\sum }_{k\in I_r}{p}_k\parallel t_{k0}(x-\xi e),z_1,\dots ,z_{n-1}\parallel =0\\ \text{for some }\xi \text{ and for every nonzero }{z}_1,\dots ,z_{n-1}\in X\end{array}\right\},\hfill \\ |C_1,\theta {|}_n=\left\{\begin{array}{l}x:{lim}_{r\to \mathrm{\infty }}\frac{1}{H_r}{\sum }_{k\in I_r}{p}_k{t}_{k0}\parallel x-\xi e,z_1,\dots ,z_{n-1}\parallel =0,\\ \text{for some }\xi \text{ and for every nonzero }{z}_1,\dots ,z_{n-1}\in X\end{array}\right\}.\hfill \end{array}$$

• (2) If we take $p_k=1$ for all $k\in \mathbb{N}$, then the sequence spaces above are reduced to the following spaces:

  $$\begin{array}{c}{[w_{\theta }]}_n=\left\{\begin{array}{l}x:{lim}_{r\to \mathrm{\infty }}\parallel \frac{1}{h_r}{\sum }_{k\in I_r}{t}_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel =0,\text{ uniformly in }m,\\ \text{for some }\xi \text{ and for every nonzero }{z}_1,\dots ,z_{n-1}\in X\end{array}\right\},\hfill \\ {(w_{\theta })}_n=\left\{\begin{array}{l}x:{lim}_{r\to \mathrm{\infty }}\frac{1}{h_r}{\sum }_{k\in I_r}\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel =0,\text{ uniformly in }m,\\ \text{for some }\xi \text{ and for every nonzero }{z}_1,\dots ,z_{n-1}\in X\end{array}\right\},\hfill \\ |w_{\theta }{|}_n=\left\{\begin{array}{l}x:{lim}_{r\to \mathrm{\infty }}\frac{1}{h_r}{\sum }_{k\in I_r}{t}_{km}(\parallel x-\xi e,z_1,\dots ,z_{n-1}\parallel )=0,\text{ uniformly in }m,\\ \text{for some }\xi \text{ and for every nonzero }{z}_1,\dots ,z_{n-1}\in X\end{array}\right\}.\hfill \end{array}$$

• (3) Let us choose $\theta =(k_r)=2^r$ for $r>0$, then these sequence spaces above are reduced to the following spaces:

  $$\begin{array}{c}{[\tilde{R},p_r]}_n=\left\{\begin{array}{l}x:{lim}_{r\to \mathrm{\infty }}\parallel \frac{1}{P_r}{\sum }_{k=1}^r{p}_k{t}_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel =0,\text{ uniformly}\\ \text{in }m,\text{ for some }\xi \text{ and for every nonzero }{z}_1,\dots ,z_{n-1}\in X\end{array}\right\},\hfill \\ {(\tilde{R},p_r)}_n=\left\{\begin{array}{l}x:{lim}_{r\to \mathrm{\infty }}\frac{1}{P_r}{\sum }_{k=1}^r{p}_k\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel =0,\text{ uniformly}\\ \text{in }m,\text{ for some }\xi \text{ and for every nonzero }{z}_1,\dots ,z_{n-1}\in X\end{array}\right\},\hfill \\ |\tilde{R},p_r{|}_n=\left\{\begin{array}{l}x:{lim}_{r\to \mathrm{\infty }}\frac{1}{P_r}{\sum }_{k=1}^r{p}_k{t}_{km}(\parallel x-\xi e,z_1,\dots ,z_{n-1}\parallel )=0,\text{ uniformly}\\ \text{in }m,\text{ for some }\xi \text{ and for every nonzero }{z}_1,\dots ,z_{n-1}\in X\end{array}\right\}.\hfill \end{array}$$

• (4) If we select $\theta =(k_r)=2^r$ for $r>0$ and the base space as $(X,\parallel \cdot ,\cdot \parallel )$ then these sequence spaces above are reduced to the sequence spaces which can be seen in [5].

• (5) If we choose $p_k=1$ for all $k\in \mathbb{N}$ and $\theta =(k_r)=2^r$ for $r>0$, then these sequence spaces above are reduced to the sequence spaces ${[C_1]}_n$, ${(C_1)}_n$, $|C_1{|}_n$, respectively.

Now, we give the following theorem to demonstrate some inclusion relations among the sequence spaces $|\tilde{R},p_r,\theta {|}_n$, ${(\tilde{R},p_r,\theta )}_n$, ${[\tilde{R},p_r,\theta ]}_n$, $|C_1,\theta {|}_n$, ${(C_1,\theta )}_n$, ${[C_1,\theta ]}_n$ with the spaces F and $[F]$.

Theorem 3.1 The following statements are true:

• (1) $[F]\subset F\subset {(\tilde{R},p_r,\theta )}_n\subset {[\tilde{R},p_r,\theta ]}_n\subset {[C_1,\theta ]}_n$.

• (2) $[F]\subset |\tilde{R},p_r,\theta {|}_n\subset {(\tilde{R},p_r,\theta )}_n\subset {[\tilde{R},p_r,\theta ]}_n\subset {[C_1,\theta ]}_n$.

• (3) $[F]\subset |\tilde{R},p_r,\theta {|}_n\subset |C_1,\theta |\subset {(C_1,\theta )}_n\subset {[C_1,\theta ]}_n$.

Proof We give the proof only for (2). The proofs of (1) and (3) can be done, similarly. So we omit them. Let $x\in [F]$ and $[F]-limx=\xi$. Then $t_{km}(\parallel x-\xi e,z_1,\dots ,z_{n-1}\parallel )\to 0$ as $k\to \mathrm{\infty }$, uniformly in m, for every nonzero $z_1,\dots ,z_{n-1}\in X$. Since $H_r\to \mathrm{\infty }$ as $r\to \mathrm{\infty }$, then its weighted lacunary mean also converges to ξ as $r\to \mathrm{\infty }$ uniformly in m. This proves that $x\in |\tilde{R},p_r,\theta {|}_n$ and $[F]-limx=|\tilde{R},p_r,\theta {|}_n-limx=\xi$. Also since

$$\begin{array}{rl}\parallel \frac{1}{H_r}\sum _{k\in I_r}{p}_k{t}_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel & ⩽\frac{1}{H_r}\sum _{k\in I_r}{p}_k\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel \\ ⩽\frac{1}{H_r}\sum _{k\in I_r}{p}_k{t}_{km}(\parallel x-\xi e,z_1,\dots ,z_{n-1}\parallel ),\end{array}$$

then it follows that $[F]\subset |\tilde{R},p_r,\theta {|}_n\subset {(\tilde{R},p_r,\theta )}_n\subset {[\tilde{R},p_r,\theta ]}_n$ and $[F]-limx=|\tilde{R},p_r,\theta {|}_n-limx={(\tilde{R},p_r,\theta )}_n-limx={[\tilde{R},p_r,\theta ]}_n-limx=\xi$. Since uniform convergence of $\parallel \frac{1}{H_r}{\sum }_{k\in I_r}{p}_k{t}_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel$ with respect to m, as $r\to \mathrm{\infty }$, implies convergence for $m=0$ and for every nonzero $z_1,\dots ,z_{n-1}\in X$. It follows that ${[\tilde{R},p_r,\theta ]}_n\subset {[C_1,\theta ]}_n$ and ${[\tilde{R},p_r,\theta ]}_n-limx={[C_1,\theta ]}_n-limx=\xi$. This completes the proof. □

Theorem 3.2 Let $\theta =(k_r)$ be a lacunary sequence and ${lim\hspace{0.17em}inf}_r{Q}_r>1$. Then ${(\tilde{R},p_r)}_n\subseteq {(\tilde{R},p_r,\theta )}_n$ with ${(\tilde{R},p_r)}_n-limx={(\tilde{R},p_r,\theta )}_n-limx=\xi$.

Proof Suppose that ${lim\hspace{0.17em}inf}_r{Q}_r>1$, then there exists a $\delta >0$ such that $Q_r⩾1+\delta$ for sufficiently large values of r, which implies that $\frac{H_r}{P_{k_r}}⩾\frac{\delta }{1+\delta }$. If $x\in {(\tilde{R},p_r)}_n$ with ${(\tilde{R},p_r)}_n-limx=\xi$, then for sufficiently large values of r, we have

$$\begin{array}{r}\frac{1}{P_{k_r}}\sum _{k=1}^{k_r}{p}_k\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel \\ =\frac{1}{P_{k_r}}(\sum _{k=1}^{k_{r-1}}{p}_k\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel +\sum _{k=k_{r-1}+1}^{k_r}{p}_k\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel )\\ ⩾\frac{H_r}{P_{k_r}}\left(\frac{1}{H_r}\sum _{k\in I_r}{p}_k\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel \right)\\ ⩾\frac{\delta }{1+\delta }.\frac{1}{H_r}\sum _{k\in I_r}{p}_k\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel ,\end{array}$$

for each $m⩾0$ and for every nonzero $z_1,\dots ,z_{n-1}\in X$. Then, it follows that $x\in {(\tilde{R},p_r,\theta )}_n$ with ${(\tilde{R},p_r,\theta )}_n-limx=\xi$ by taking the limit as $r\to \mathrm{\infty }$. This completes the proof. □

Theorem 3.3 Let $\theta =(k_r)$ be a lacunary sequence with ${lim\hspace{0.17em}sup}_r{Q}_r<\mathrm{\infty }$. Then ${(\tilde{R},p_r,\theta )}_n\subseteq {(\tilde{R},p_r)}_n$ with ${(\tilde{R},p_r,\theta )}_n-limx={(\tilde{R},p_r)}_n-limx=\xi$.

Proof Let $x\in {(\tilde{R},p_r,\theta )}_n$ with ${(\tilde{R},p_r,\theta )}_n-limx=\xi$. Then for $\epsilon >0$, there exists $q_0$ such that for every $q>q_0$

$$L_q=\frac{1}{H_q}\sum _{k\in I_q}{p}_k\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel <\epsilon ,$$

(3.1)

for each $m⩾0$ and for every nonzero $z_1,\dots ,z_{n-1}\in X$, that is, we can find some positive constant M such that

$$L_q⩽M\text{for all }q.$$

(3.2)

${lim\hspace{0.17em}sup}_r{Q}_r<\mathrm{\infty }$ implies that there exists some positive number K such that

$$Q_r⩽K\text{for all }r⩾1.$$

(3.3)

Therefore for $k_{r-1}<r⩽k_r$, we have by (3.1), (3.2), and (3.3)

$$\begin{array}{r}\frac{1}{P_r}\sum _{k=1}^r{p}_k\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel \\ ⩽\frac{1}{P_{k_{r-1}}}\sum _{k=1}^{k_r}{p}_k\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel \\ =\frac{1}{P_{k_{r-1}}}(\sum _{k\in I_1}{p}_k\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel +\sum _{k\in I_2}{p}_k\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel +\cdots \\ +\sum _{k\in I_{q_0}}{p}_k\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel +\cdots +\sum _{k\in I_r}{p}_k\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel )\\ =\frac{1}{P_{k_{r-1}}}(L_1{H}_1+L_2{H}_2+\cdots +L_{q_0}{H}_{q_0}+L_{q_0+1}{H}_{q_0+1}+\cdots +L_r{H}_r)\\ ⩽\frac{M}{P_{k_{r-1}}}(H_1+H_2+\cdots +H_{q_0})+\frac{\epsilon }{P_{k_{r-1}}}(H_{q_0+1}+\cdots +H_r)\\ =\frac{M}{P_{k_{r-1}}}(P_{k_1}-P_{k_0}+\cdots +P_{k_{q_0}}-P_{k_{q_0-1}})+\frac{\epsilon }{P_{k_{r-1}}}(P_{k_{q_0}}-P_{k_{q_0-1}}+\cdots +P_{k_r}-P_{k_{r-1}})\\ =M\frac{P_{k_{q_0}}}{P_{k_{r-1}}}+\epsilon \frac{P_{k_r}-P_{k_{q_0}}}{P_{k_{r-1}}}\\ ⩽M\frac{P_{k_{q_0}}}{P_{k_{r-1}}}+\epsilon K,\end{array}$$

for each $m⩾0$ and for every nonzero $z_1,\dots ,z_{n-1}\in X$. Since $P_{k_{r-1}}\to \mathrm{\infty }$ as $r\to \mathrm{\infty }$, we get $x\in {(\tilde{R},p_r)}_n$ with ${(\tilde{R},p_r)}_n-limx=\xi$. This completes the proof. □

Corollary 3.4 Let $1<{lim\hspace{0.17em}inf}_r{Q}_r⩽{lim\hspace{0.17em}sup}_r{Q}_r<\mathrm{\infty }$. Then ${(\tilde{R},p_r,\theta )}_n={(\tilde{R},p_r)}_n$ and ${(\tilde{R},p_r,\theta )}_n-limx={(\tilde{R},p_r)}_n-limx=\xi$.

Proof It follows from Theorem 3.2 and Theorem 3.3. □

In the following theorem, we give the relations between the sequence spaces ${(w_{\theta })}_n$ and ${(\tilde{R},p_r)}_n$.

Theorem 3.5

• (1) If $p_k<1$ for all $k\in \mathbb{N}$, then ${(w_{\theta })}_n\subseteq {(\tilde{R},p_r)}_n$ and ${(w_{\theta })}_n-limx={(\tilde{R},p_r)}_n-limx=\xi$.

• (2) If $p_k>1$ for all $k\in \mathbb{N}$ and $(\frac{H_r}{h_r})$ is upper-bounded, then ${(\tilde{R},p_r)}_n\subseteq {(w_{\theta })}_n$ and ${(\tilde{R},p_r)}_n-limx={(w_{\theta })}_n-limx=\xi$.

Proof

• (1) If $p_k<1$ for all $k\in \mathbb{N}$, then $H_r<h_r$ for all $r\in \mathbb{N}$. So, there exists an $M_1$, a constant, such that $0<M_1⩽\frac{H_r}{h_r}<1$ for all $r\in \mathbb{N}$. Let $x\in {(w_{\theta })}_n$ with ${(w_{\theta })}_n-limx=\xi$, then for an arbitrary $\epsilon >0$ we have

  $$\frac{1}{H_r}\sum _{k\in I_r}{p}_k\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel ⩽\frac{1}{M_1}\frac{1}{h_r}\sum _{k\in I_r}\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel ,$$

  for each $m⩾0$ and for every nonzero $z_1,\dots ,z_{n-1}\in X$. Therefore, we get the result by taking the limit as $r\to \mathrm{\infty }$.

• (2) Let $p_k>1$ for all $k\in \mathbb{N}$, then $H_r>h_r$ for all $r\in \mathbb{N}$. Suppose that $(\frac{H_r}{h_r})$ is upper-bounded, then there exists an $M_2$, a constant, such that $1<\frac{H_r}{h_r}⩽M_2<\mathrm{\infty }$ for all $r\in \mathbb{N}$. Let $x\in {(\tilde{R},p_r)}_n$ and ${(\tilde{R},p_r)}_n-limx=\xi$. So the result is obtained by taking the limit as $r\to \mathrm{\infty }$ for each $m⩾0$ and for every nonzero $z_1,\dots ,z_{n-1}\in X$, from the following inequality:

  $$\frac{1}{h_r}\sum _{k\in I_r}\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel ⩽M_2\frac{1}{H_r}\sum _{k\in I_r}{p}_k\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel .$$

 □

Now, we define a new concept of statistical convergence in n-normed space, which will be called weighted almost lacunary statistical convergence:

Definition 3.6 The weighted almost lacunary density of $K\subseteq \mathbb{N}$ is denoted by ${\delta }_{(\tilde{R},\theta )}(K)={lim}_{r\to \mathrm{\infty }}\frac{1}{H_r}|K_r(\epsilon )|$ if the limit exists. We say that the sequence $x=(x_j)$ is weighted almost lacunary statistically convergent to ξ if for every $\epsilon >0$, the set $K_r(\epsilon )=\{k\in I_r^{\prime }:p_k\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel ⩾\epsilon \}$ has weighted lacunary density zero, i.e.

$$\underset{r\to \mathrm{\infty }}{lim}\frac{1}{H_r}\left|\{k\in I_r^{\prime }:p_k\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel ⩾\epsilon \}\right|=0$$

(3.4)

uniformly in m, for every nonzero $z_1,\dots ,z_{n-1}\in X$. In this case, we write $(S_{(\tilde{R},\theta )},n)-{lim}_k{x}_k=\xi$. By $(S_{(\tilde{R},\theta )},n)$ we denote the set of all weighted almost lacunary statistically convergent sequences in n-normed space.

• (1) If we take $p_k=1$ for all $k\in \mathbb{N}$ in (3.4) then we obtain the definition of almost lacunary statistical convergence in n−normed space, that is, x is called almost lacunary statistically convergent to ξ if for every $\epsilon >0$, the set $K_{\theta }(\epsilon )=\{k\in I_r:\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel ⩾\epsilon \}$ has lacunary density zero, i.e.

  $$\underset{r\to \mathrm{\infty }}{lim}\frac{1}{h_r}\left|\{k\in I_r:\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel ⩾\epsilon \}\right|=0$$

  (3.5)

  uniformly in m, for every nonzero $z_1,\dots ,z_{n-1}\in X$. In this case, we write $(S_{\theta },n)-{lim}_j{x}_j=\xi$. By $(S_{\theta },n)$ we denote the set of all weighted almost lacunary statistically convergent sequences in n-normed space.

• (2) Let us choose $\theta =(k_r)$ for $r>0$ then the definition of weighted almost lacunary statistical convergence which is given in (3.4) is reduced to the definition of weighted almost statistically convergence, that is, x is called weighted almost statistically convergent to ξ if for every $\epsilon >0$, the set $K_{P_r}(\epsilon )=\{k⩽P_r:p_k\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel ⩾\epsilon \}$ has weighted density zero, i.e.

  $$\underset{r\to \mathrm{\infty }}{lim}\frac{1}{P_r}\left|\{k⩽P_r:\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel ⩾\epsilon \}\right|=0$$

  (3.6)

  uniformly in m, for every nonzero $z_1,\dots ,z_{n-1}\in X$. In this case, we write $(S_{\tilde{R}},n)-{lim}_j{x}_j=\xi$. By $(S_{\tilde{R}},n)$ we denote the set of all weighted almost lacunary statistically convergent sequences in n-normed space.

• (3) Let us choose $\theta =(k_r)$ for $r>0$ and $p_k=1$ for all $k\in \mathbb{N}$, then the definition of weighted almost lacunary statistical convergence, which is given in (3.4), is reduced to the definition of almost statistical convergence.

Theorem 3.7 If the sequence x is ${(\tilde{R},p_r,\theta )}_n$-convergent to ξ then the sequence x is weighted almost lacunary statistically convergent to ξ.

Proof Let the sequence x be ${(\tilde{R},p_r,\theta )}_n$-convergent to ξ and $K_{rm}(\epsilon )=\{k\in I_r^{\prime }:p_k\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel ⩾\epsilon \}$. Then for a given $\epsilon >0$, we have

$$\begin{array}{rl}\frac{1}{H_r}\sum _{k\in I_r}{p}_k\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel & ⩾\frac{1}{H_r}\underset{k\in K_{rm}(\epsilon )}{\sum _{k\in I_r}}{p}_k\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel \\ ⩾\epsilon \frac{1}{H_r}|K_{rm}(\epsilon )|\end{array}$$

for each $m\ge 0$ and for every nonzero $z\in X$. Hence, we see that the sequence x is weighted almost statistically convergent to ξ by taking the limit as $r\to \mathrm{\infty }$. □

Theorem 3.8 Let $p_k\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel ⩽M$ for all $k\in \mathbb{N}$, for each $m⩾0$ and for every nonzero $z_1,\dots ,z_{n-1}\in X$. Then $(S_{(\tilde{R},\theta )},n)\subset {(\tilde{R},p_r,\theta )}_n$ with $(S_{(\tilde{R},\theta )},n)-limx={(\tilde{R},p_r,\theta )}_n-limx=\xi$.

Proof Let x be convergent to ξ in $(S_{(\tilde{R},\theta )},n)$ and let us take

$$K_{rm}(\epsilon )=\{k\in I_r^{\prime }:p_k\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel ⩾\epsilon \}.$$

Since $p_k\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel ⩽M$ for all $k\in \mathbb{N}$ for each $m\ge 0$, for every nonzero $z_1,\dots ,z_{n-1}\in X$ and $H_r\to \mathrm{\infty }$ as $r\to \mathrm{\infty }$, then for a given $\epsilon >0$ we have

$$\begin{array}{rl}\frac{1}{H_r}\sum _{k\in I_r}{p}_k\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel =& \frac{1}{H_r}\underset{k\in K_{rm(\epsilon )}}{\sum _{k\in I_r}}{p}_k\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel \\ +\frac{1}{H_r}\underset{k\notin K_{rm(\epsilon )}}{\sum _{k\in I_r}}{p}_k\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel \\ ⩽& M\frac{1}{H_r}|K_{rm(\epsilon )}|+\frac{h_r}{H_r}\epsilon \\ ⩽& M\frac{1}{H_r}|K_{rm(\epsilon )}|+\epsilon ,\end{array}$$

for each $m\ge 0$ and for every nonzero $z_1,\dots ,z_{n-1}\in X$. Since ε is arbitrary, we have $x\in {(\tilde{R},p_r,\theta )}_n$ by taking the limit as $r\to \mathrm{\infty }$. □

Theorem 3.9 The following statements are true.

• (1) If $p_k⩽1$ for all $k\in \mathbb{N}$, then $(S_{\theta },n)\subseteq (S_{(\tilde{R},\theta )},n)$.

• (2) Let $p_k⩾1$ for all $k\in \mathbb{N}$ and $(\frac{H_r}{h_r})$ be upper-bounded, then $(S_{(\tilde{R},\theta )},n)\subseteq (S_{\theta },n)$.

Proof

• (1) If $p_k⩽1$ for all $k\in \mathbb{N}$, then $H_r⩽h_r$ for all $r\in \mathbb{N}$. So, there exist $M_1$ and $M_2$, constants, such that $0<M_1⩽\frac{H_r}{h_r}⩽M_2⩽1$ for all $r\in \mathbb{N}$. Let $x\in (S_{\theta },n)$ with $(S_{\theta },n)-limx=\xi$, then for an arbitrary $\epsilon >0$ we have

  $$\begin{array}{r}\frac{1}{H_r}\left|\{k\in I_r^{\prime }:p_k\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel ⩾\epsilon \}\right|\\ =\frac{1}{H_r}\left|\{P_{k_{r-1}}<k⩽P_{k_r}:p_k\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel ⩾\epsilon \}\right|\\ ⩽\frac{1}{M_1}\frac{1}{h_r}\left|\{P_{k_{r-1}}⩽k_{r-1}<k⩽P_{k_r}⩽k_r:\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel ⩾\epsilon \}\right|\\ =\frac{1}{M_1}\frac{1}{h_r}\left|\{k_{r-1}<k⩽k_r:\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel ⩾\epsilon \}\right|\\ =\frac{1}{M_1}\frac{1}{h_r}\left|\{k\in I_r:\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel ⩾\epsilon \}\right|,\end{array}$$

for each $m⩾0$ and for every nonzero $z_1,\dots ,z_{n-1}\in X$. Hence, we obtain the result by taking the limit as $r\to \mathrm{\infty }$.

• (2) Let $(\frac{H_r}{h_r})$ be upper-bounded, then there exist $M_1$ and $M_2$, constants, such that $1⩽M_1⩽\frac{H_r}{h_r}⩽M_2<\mathrm{\infty }$ for all $r\in \mathbb{N}$. Suppose that $p_k⩾1$ for all $k\in \mathbb{N}$, then $H_r⩾h_r$ for all $r\in \mathbb{N}$. Let $x\in {(\tilde{R},p_r)}_n$ and ${(\tilde{R},p_r)}_n-limx=\xi$, then for an arbitrary $\epsilon >0$ we have

  $$\begin{array}{r}\frac{1}{h_r}\left|\{k\in I_r:\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel ⩾\epsilon \}\right|\\ =\frac{1}{h_r}\left|\{k_{r-1}<k⩽k_r:\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel ⩾\epsilon \}\right|\\ ⩽M_2\frac{1}{H_r}\left|\{k_{r-1}⩽P_{k_{r-1}}<k⩽k_r⩽P_{k_r}:p_k\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel ⩾\epsilon \}\right|\\ =M_2\frac{1}{H_r}\left|\{P_{k_{r-1}}<k⩽P_{k_r}:p_k\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel ⩾\epsilon \}\right|\\ =M_2\frac{1}{H_r}\left|\{k\in I_r^{\prime }:p_k\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel ⩾\epsilon \}\right|,\end{array}$$

for each $m⩾0$ and for every nonzero $z_1,\dots ,z_{n-1}\in X$. Hence, the result is obtained by taking the limit as $r\to \mathrm{\infty }$. □

Theorem 3.10 For any lacunary sequence θ, if $lim{inf}_r{Q}_r>1$ then $(S_{\tilde{R}},n)\subseteq (S_{(\tilde{R},\theta )},n)$ and $(S_{\tilde{R}},n)-limx=(S_{(\tilde{R},\theta )},n)-limx=\xi$.

Proof Suppose that $lim{inf}_r{Q}_r>1$, then there exists a $\delta >0$ such that $Q_r⩾1+\delta$ for sufficiently large values of r, which implies that $\frac{H_r}{P_{k_r}}⩾\frac{\delta }{1+\delta }$. If $x\in (S_{\tilde{R}},n)$ with $(S_{\tilde{R}},n)-limx=\xi$, then for every $\epsilon >0$ and for sufficiently large values of r, we have

$$\begin{array}{r}\frac{1}{P_{k_r}}\left|\{k⩽P_{k_r}:p_k\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel ⩾\epsilon \}\right|\\ ⩾\frac{1}{P_{k_r}}\left|\{P_{k_{r-1}}<k⩽P_{k_r}:p_k\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel ⩾\epsilon \}\right|\\ =\frac{H_r}{P_{k_r}}\left(\frac{1}{H_r}\right|\{P_{k_{r-1}}<k⩽P_{k_r}:p_k\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel ⩾\epsilon \}\left|\right)\\ ⩾\frac{\delta }{1+\delta }\left(\frac{1}{H_r}\right|\{k\in I_r^{\prime }:p_k\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel ⩾\epsilon \}\left|\right),\end{array}$$

for each $m⩾0$ and for every nonzero $z_1,\dots ,z_{n-1}\in X$. Hence, we get the result by taking the limit as $r\to \mathrm{\infty }$. □

Theorem 3.11 Let $\theta =(k_r)$ be a lacunary sequence with $lim{sup}_r{Q}_r<\mathrm{\infty }$, then $(S_{(\tilde{R},\theta )},n)\subseteq (S_{\tilde{R}},n)$ and $(S_{\tilde{R}},n)-limx=(S_{(\tilde{R},\theta )},n)-limx=\xi$.

Proof If $lim{sup}_r{Q}_r<\mathrm{\infty }$, then there is a $K>0$ such that $Q_r⩽K$ for all $r\in \mathbb{N}$. Suppose that $x\in (S_{(\tilde{R},\theta )},n)$ with $(S_{(\tilde{R},\theta )},n)-limx=\xi$ and let

$$N_r:=\left|\{k\in I_r^{\prime }:p_k\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel ⩾\epsilon \}\right|.$$

(3.7)

By (3.7), given $\epsilon >0$, there is a $r_0\in \mathbb{N}$ such that $\frac{N_r}{H_r}<\epsilon$ for all $r>r_0$. Now, let $M:=max\{N_r:1⩽r⩽r_0\}$ and let r be any integer satisfying $k_{r-1}<r⩽k_r$, then for each $m⩾0$ and for every nonzero $z_1,\dots ,z_{n-1}\in X$ we can write

$$\begin{array}{r}\frac{1}{P_r}\left|\{k⩽P_r:p_k\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel ⩾\epsilon \}\right|\\ ⩽\frac{1}{P_{k_{r-1}}}\left|\{P_{k_{r-1}}<k⩽P_{k_r}:p_k\parallel t_{km}(x-\xi e),z_1,\dots ,z_{n-1}\parallel ⩾\epsilon \}\right|\\ =\frac{1}{P_{k_{r-1}}}(N_1+N_2+\cdots +N_{r_0}+N_{r_0+1}+\cdots +N_r)\\ ⩽\frac{M.r_0}{P_{k_{r-1}}}+\frac{1}{P_{k_{r-1}}}\epsilon (H_{r_0+1}+\cdots +H_r)\\ =\frac{M.r_0}{P_{k_{r-1}}}+\epsilon \frac{(P_{k_r}-P_{k_{r_0}})}{P_{k_{r-1}}}\\ ⩽\frac{M.r_0}{P_{k_{r-1}}}+\epsilon Q_r⩽\frac{M.r_0}{P_{k_{r-1}}}+\epsilon K,\end{array}$$

which completes the proof by taking the limit as $r\to \mathrm{\infty }$. □

Corollary 3.12 Let $1<lim{inf}_r{Q}_r⩽lim{sup}_r{Q}_r<\mathrm{\infty }$. Then $(S_{(\tilde{R},\theta )},n)=(S_{\tilde{R}},n)$ and $(S_{\tilde{R}},n)-limx=(S_{(\tilde{R},\theta )},n)-limx=\xi$.

Proof It follows from Theorem 3.10 and Theorem 3.11. □