Some new lacunary statistical convergence with ideals

Abstract

In this paper, the idea of lacunary $I_{\lambda }$-statistical convergent sequence spaces is discussed which is defined by a Musielak-Orlicz function. We study relations between lacunary $I_{\lambda }$-statistical convergence with lacunary $I_{\lambda }$-summable sequences. Moreover, we study the $I_{\lambda }$-lacunary statistical convergence in probabilistic normed space and discuss some topological properties.

Introduction

The concept of statistical convergence [1] which is the extended idea of convergence of real sequences has become an important tool in many branches of mathematics. For references one may see [2–8] and many more.

Similarly, I-convergence is also an extended notion of statistical convergence ([9]) of real sequences. A family of sets $I\subseteq 2^A$ (power sets of A) is an ideal if I is additive, i.e. $S,T\in I\Rightarrow S\cup T\in I$, and hereditary i.e. $S\in I$, $T\subseteq S\Rightarrow T\in I$, where A is any non-empty set.

A lacunary sequence is an increasing integer sequence $\theta =(i_j)$ such that $i_0=0$ and $h_j=i_j-i_{j-1}\to \mathrm{\infty }$ as $j\to \mathrm{\infty }$. As regards ideal convergence and lacunary ideal convergence, one may refer to [10–19] etc.

Note: Throughout this paper, θ will be determined by the interval $K_j=(k_{j-1},k_j]$ and the ratio $\frac{k_j}{k_{j-1}}$ will be defined by ${\varphi }_j$.

Preliminary concepts

A sequence $(x_i)$ of real numbers is statistically convergent to M if, for arbitrary $\xi >0$, the set $K(\xi )=\{i\in \mathbb{N}:|x_i-M|\ge \xi \}$ has natural density zero, i.e.,

$$\underset{i}{lim}\frac{1}{i}\sum _{j=1}^i{\chi }_{K(\xi )}(j)=0,$$

where ${\chi }_{K(\xi )}$ denotes the characteristic function of $K(\xi )$.

A sequence $(x_i)$ of elements of $\mathbb{R}$ is I-convergent to $M\in \mathbb{R}$ if, for each $\xi >0$,

$$\{i\in \mathbb{N}:|x_i-M|\ge \xi \}\in I.$$

For any lacunary sequence $\theta =(i_j)$, the space $N_{\theta }$ is defined as (Freedman et al. [5])

$$N_{\theta }=\{(x_i):\underset{j\to \mathrm{\infty }}{lim}{i}_j^{-1}\sum _{i\in K_j}|x_i-M|=0,\text{ for some }M\}.$$

The concept of a Musielak-Orlicz function is defined as $\mathcal{M}=(M_j)$. The sequence $\mathcal{N}=(N_i)$ is defined by

$$N_i(a)=sup\{|a|b-M_j(b):b\ge 0\},i=1,2,\dots ,$$

which is named the complementary function of a Musielak-Orlicz function $\mathcal{M}$ (see [20]) (throughout the paper $\mathcal{M}$ is a Musielak-Orlicz function).

If $\lambda =({\lambda }_i)$ is a non-decreasing sequence of positive integers such that Λ denotes the set of all non-decreasing sequences of positive integers. We call a sequence ${\{x_i\}}_{i\in \mathbb{N}}$ lacunary $I_{\lambda }$-statistically convergent of order α to M, if, for each $\gamma >0$ and $\xi >0$,

$$\{i\in \mathbb{N}:\frac{1}{{\lambda }_i^{\alpha }}|\{j\le i:\frac{1}{h_i}\sum _{j\in I_i}{M}_j\left(\frac{|x_j-M|}{{\rho }^{(j)}}\right)\ge \gamma \}|\ge \xi \}\in I.$$

We denote the class of all lacunary $I_{\lambda }$-statistically convergent sequences of order α defined by a Musielak-Orlicz function by $S_{I_{\lambda }}^{\alpha }(\mathcal{M},\theta )$.

Some particular cases:

1. If $M_j(x)=M(x)$, for all $j\in \mathbb{N}$, then $S_{I_{\lambda }}^{\alpha }(\mathcal{M},\theta )$ is reduced to $S_{I_{\lambda }}^{\alpha }(M,\theta )$.

   Also, if $M_j(x)=x$, for all $j\in \mathbb{N}$, then $S_{I_{\lambda }}^{\alpha }(\mathcal{M},\theta )$ will be changed as $S_{I_{\lambda }}^{\alpha }(\theta )$.

2. If ${\lambda }_i=i$, for all $i\in \mathbb{N}$, then $S_{I_{\lambda }}^{\alpha }(\mathcal{M},\theta )$ will be reduced to $S_I^{\alpha }(\mathcal{M},\theta )$.

3. If $\alpha =1$, then α-density of any set is reduced to the natural density of the set. So, the set $S_{I_{\lambda }}^{\alpha }(\mathcal{M},\theta )$ reduces to $S_{I_{\lambda }}(\mathcal{M},\theta )$ for $\alpha =1$.

4. If $\theta =(2^r)$ and $\alpha =1$, then $(x_j)$ is said to be $I_{\lambda }$-statistically convergent defined by a Musielak-Orlicz function, i.e. $(x_j)\in S_{I_{\lambda }}(\mathcal{M})$.

5. if $M_j(x)=x$, $\theta =(2^r)$, ${\lambda }_j=j$, $\alpha =1$, then $I_{\lambda }$-lacunary statistically convergence of order α defined by Musielak-Orlicz function reduces to I-statistical convergence.

In this article, we define the concept of lacunary $I_{\lambda }$-statistically convergence of order α defined by $\mathcal{M}$ and investigate some results on these sequences. Later on, we investigate some results of lacunary $I_{\lambda }$-statistically convergence of real sequences in probabilistic normed space too.

Main results

Theorem 3.1

Let $\lambda =({\lambda }_i)$ and $\mu =({\mu }_i)$ be two sequences in Λ such that ${\lambda }_i\le {\mu }_i$ for all $i\in \mathbb{N}$ and $0<\alpha \le \beta \le 1$ for fixed reals α and β. If $lim{inf}_{i\to \mathrm{\infty }}\frac{{\lambda }_i^{\alpha }}{{\mu }_i^{\beta }}>0$, then $S_{I_{\mu }}^{\beta }(\mathcal{M},\theta )\subseteq S_{I_{\lambda }}^{\alpha }(\mathcal{M},\theta )$.

Proof

Suppose that ${\lambda }_i\le {\mu }_i$ for all $i\in \mathbb{N}$ and $lim{inf}_{i\to \mathrm{\infty }}\frac{{\lambda }_i^{\alpha }}{{\mu }_i^{\beta }}>0$. Since $I_i\subset J_i$, where $J_i=[i-{\mu }_i+1,i]$, so for $\gamma >0$, we can write

$$\{j\in J_i:|x_j-M|\ge \gamma \}\supset \{j\in I_i:|x_j-M|\ge \gamma \},$$

which implies

$$\frac{1}{{\mu }_i^{\beta }}\left|\{j\in J_i:|x_j-M|\ge \gamma \}\right|\ge \frac{{\lambda }_i^{\alpha }}{{\mu }_i^{\beta }}.\frac{1}{{\lambda }_i^{\alpha }}\left|\{j\in I_i:|x_j-M|\ge \gamma \}\right|,$$

for all $i\in \mathbb{N}$.

Assume that $lim{inf}_{i\to \mathrm{\infty }}\frac{{\lambda }_i^{\alpha }}{{\mu }_i^{\beta }}=a$, so from the definition we see that $\{i\in \mathbb{B}:\frac{{\lambda }_i^{\alpha }}{{\mu }_i^{\beta }}<\frac{a}{2}\}$ is finite. Now for $\xi >0$,

$$\begin{array}{rcl}\{i\in \mathbb{N}:\frac{1}{{\lambda }_i^{\beta }}|\{j\in J_i:|x_j-M|\ge \gamma \}|\ge \xi \}& \subset & \{i\in \mathbb{N}:\frac{1}{{\mu }_i^{\alpha }}|\{j\in I_i:|x_j-M|\ge \gamma \}|\ge \frac{a}{2}\xi \}\\ & & \cup \{i\in \mathbb{N}:\frac{{\lambda }_i^{\alpha }}{{\mu }_i^{\beta }}<\frac{a}{2}\}.\end{array}$$

Since I is admissible and $(x_j)$ is a lacunary $I_{\mu }$-statistically convergent sequence of order β defined by $\mathcal{M}$, by using the continuity of $\mathcal{M}$, we see with the lacunary sequence $\theta =(h_i)$, the right hand side belongs to I, which completes the proof. □

Theorem 3.2

If ${lim}_{i\to \mathrm{\infty }}\frac{{\mu }_i}{{\lambda }_i^{\beta }}=1$, for $\lambda =({\lambda }_i)$ and $\mu =({\mu }_i)$ two sequences of Λ such that ${\lambda }_i\le {\mu }_i$, $\mathrm{\forall }i\in \mathbb{N}$ and $0<\alpha \le \beta \le 1$ for fixed α, β reals, then $S_{I_{\lambda }}^{\alpha }(\mathcal{M},\theta )\subseteq S_{I_{\mu }}^{\beta }(\mathcal{M},\theta )$.

Proof

Let $(x_j)$ be lacunary $I_{\lambda }$-statistically convergent to M of order α defined by $\mathcal{M}$. Also assume that ${lim}_{i\to \mathrm{\infty }}\frac{{\mu }_i}{{\lambda }_i^{\beta }}=1$. Choose $m\in \mathbb{N}$ such that $|\frac{{\mu }_i}{{\lambda }_i^{\beta }}-1|<\frac{\xi }{2}$, $\mathrm{\forall }i\ge m$.

Since $I_i\subset J_i$, for $\gamma >0$, we may write

$$\begin{array}{rcl}\frac{1}{{\mu }_i^{\beta }}\left|\{j\in J_i:|x_j-M|\ge \gamma \}\right|& =& \frac{1}{{\mu }_i^{\beta }}\left|\{i-{\mu }_i+1\le j\le i-{\lambda }_i:|x_j-M|\ge \gamma \}\right|\\ & & +\frac{1}{{\mu }_i^{\beta }}\left|\{j\in I_i:|x_j-M|\ge \gamma \}\right|\\ & \le & \frac{{\mu }_i-{\lambda }_i}{{\mu }_i^{\beta }}+\frac{1}{{\mu }_i^{\beta }}\left|\{j\in I_i:|x_j-M|\ge \gamma \}\right|\\ & \le & \frac{{\mu }_i-{\lambda }_i^{\beta }}{{\lambda }_i^{\beta }}+\frac{1}{{\mu }_i^{\beta }}\left|\{j\in I_i:|x_j-M|\ge \gamma \}\right|\\ & \le & (\frac{{\mu }_i}{{\lambda }_i^{\beta }}-1)+\frac{1}{{\lambda }_i^{\alpha }}\left|\{j\in I_i:|x_j-M|\ge \gamma \}\right|\\ & =& \frac{\xi }{2}+\frac{1}{{\lambda }_i^{\alpha }}\left|\{j\in I_i:|x_j-M|\ge \gamma \}\right|.\end{array}$$

Hence,

$$\begin{array}{rcl}\{i\in \mathbb{N}:\frac{1}{{\mu }_i^{\beta }}|\{j\le i:|x_j-M|\ge \gamma \}|\ge \xi \}& \subset & \{i\in \mathbb{N}:\frac{1}{{\lambda }_i^{\alpha }}|\{j\in I_i:|x_j-M|\ge \gamma \}|\ge \frac{\xi }{2}\}\\ & & \cup \{1,2,3,\dots ,m\}.\end{array}$$

Since $(x_j)$ is lacunary $I_{\lambda }$-statistically convergent sequence of order α defined by $\mathcal{M}$ and since I is admissible, by using the continuity of $\mathcal{M}$, it follows that the set on the right hand side with the lacunary sequence $\theta =(h_i)$ belongs to I and

$$S_{I_{\lambda }}^{\alpha }(\mathcal{M},\theta )\subseteq S_{I_{\mu }}^{\beta }(\mathcal{M},\theta ).$$

 □

We define the lacunary $I_{\lambda }$-summable sequence of order α defined by $\mathcal{M}$ as

$$w_{I_{\lambda }}^{\alpha }(\mathcal{M},\theta )=\{i\in \mathbb{N}:\frac{1}{{\lambda }_i^{\alpha }}(j\le i:\frac{1}{h_i}\sum _{j\in I_i}{M}_j\left(\frac{|x_j-M|}{{\rho }^{(j)}}\right)\ge \gamma )\}\in I.$$

Theorem 3.3

Given $\lambda =({\lambda }_i)$, $\mu =({\mu }_i)\in \mathrm{\Lambda }$. Suppose that ${\lambda }_i\le {\mu }_i$ for all $i\in \mathbb{N}$, $0<\alpha \le \beta \le 1$. Then:

1. If $lim{inf}_{i\to \mathrm{\infty }}\frac{{\lambda }_i^{\alpha }}{{\mu }_i^{\beta }}>0$, then $w_{\mu }^{\beta }(\mathcal{M},\theta )\subset w_{\lambda }^{\alpha }(\mathcal{M},\theta )$.

2. If ${lim}_{i\to \mathrm{\infty }}\frac{{\mu }_i}{{\lambda }_i^{\beta }}=1$, then ${\ell }_{\mathrm{\infty }}\cap w_{\lambda }^{\alpha }(\mathcal{M},\theta )\subset w_{\mu }^{\beta }(\mathcal{M},\theta )$.

Theorem 3.4

Let ${\lambda }_i\le {\mu }_i$ for all $i\in \mathbb{N}$, where $\lambda ,\mu \in \mathrm{\Lambda }$. Then, if $lim{inf}_{i\to \mathrm{\infty }}\frac{{\lambda }_i^{\alpha }}{{\mu }_i^{\beta }}>0$, and if $(x_j)$ is lacunary $I_{\mu }$-summable of order β defined by $\mathcal{M}$, then it is lacunary $I_{\lambda }$-statistically convergent of order α defined by $\mathcal{M}$. Here $0<\alpha \le \beta \le 1$, for fixed reals α and β.

Proof

For any $\gamma >0$, we have

$$\begin{array}{rcl}\sum _{j\in J_i}|x_j-M|& =& \sum _{j\in J_i,|x_j-M|\ge \epsilon }|x_j-M|+\sum _{j\in J_i,|x_j-M|<\epsilon }|x_j-M|\\ & \ge & \sum _{j\in I_i,|x_j-M|\ge \epsilon }|x_j-M|+\sum _{j\in I_i,|x_j-M|\ge \epsilon }|x_j-M|\\ & \ge & \sum _{j\in I_i,|x_j-M|\ge \epsilon }|x_j-M|\\ & \ge & \left|\{j\in I_i:|x_j-M|\ge \gamma \}\right|.\gamma .\end{array}$$

Therefore,

$$\begin{array}{rcl}\frac{1}{{\mu }_i^{\beta }}\sum _{j\in J_i}|x_j-M|& \ge & \frac{1}{{\mu }_i^{\beta }}\left|\{j\in I_i:|x_j-M|\ge \gamma \}\right|.\gamma \\ & \ge & \frac{{\lambda }_i^{\alpha }}{{\mu }_i^{\beta }}.\frac{1}{{\lambda }_i^{\alpha }}\left|\{j\in I_i:|x_j-M|\ge \gamma \}\right|.\gamma .\end{array}$$

If $lim{inf}_{i\to \mathrm{\infty }}\frac{{\lambda }_i^{\alpha }}{{\mu }_i^{\beta }}=a$, then $\{i\in \mathbb{N}:\frac{{\lambda }_i^{\alpha }}{{\mu }_i^{\beta }}<\frac{a}{2}\}$ is finite. So, for $\delta >0$, we get

$$\begin{array}{c}\{i\in \mathbb{N}:\frac{1}{{\lambda }_i^{\alpha }}|\{j\le i:\sum _{j\in J_i}|x_j-M|\ge \gamma \}|\ge \xi \}\hfill \\ \subset \{i\in \mathbb{N}:\frac{1}{{\mu }_i^{\beta }}\{j\in I_i:|x_j-M|\ge \gamma \}\ge \frac{a}{2}\xi \}\hfill \\ \cup \{i\in \mathbb{N}:\frac{{\lambda }_i^{\alpha }}{{\mu }_i^{\beta }}<\frac{a}{2}\}.\hfill \end{array}$$

Since I is admissible and $(x_j)$ is lacunary $I_{\mu }$-summable sequence of order β defined by $\mathcal{M}$, using its continuity and using the lacunary sequence $\theta =(h_i)$, we can conclude that $w_{I_{\mu }}^{\beta }(\mathcal{M},\theta )\subseteq S_{I_{\lambda }}^{\alpha }(\mathcal{M},\theta )$. □

Theorem 3.5

Let ${lim}_{i\to \mathrm{\infty }}\frac{{\mu }_i}{{\lambda }_i^{\beta }}=1$, where $0<\alpha \le \beta \le 1$ for fixed reals α and β and ${\lambda }_i\le {\mu }_i$, for all $i\in \mathbb{N}$, where $\lambda ,\mu \in \mathrm{\Lambda }$. Also let θ! be a refinement of θ. Let $(x_j)$ to be a bounded sequence. If $(x_j)$ is lacunary $I_{\lambda }$-statistically convergent sequence of order α defined by $\mathcal{M}$, then it is also a lacunary $I_{\mu }$-summable sequence of order β defined by $\mathcal{M}$. i.e. $S_{I_{\lambda }}^{\alpha }(\mathcal{M},\theta )\subseteq w_{I_{\mu }}^{\beta }(\mathcal{M},\theta !)$.

Proof

Suppose that $(x_j)$ is lacunary $I_{\lambda }$-statistically convergent sequence of order α defined by $\mathcal{M}$.

Given that ${lim}_{i\to \mathrm{\infty }}\frac{{\mu }_i}{{\lambda }_i^{\beta }}=1$, we can choose $s\in \mathbb{N}$ such that $|\frac{{\mu }_i}{{\lambda }_i^{\beta }}-1|<\frac{\delta }{2}$, $\mathrm{\forall }i\ge s$.

Assume that there are a finite number of points $\theta !=(j_i^{!})$ in the interval $I_i=(j_{i-1},j_i]$. Let there exists exactly one point $j_i^{!}$ of θ! in the interval $I_i$, that is, $j_{i-1}=j_{p-1}^{!}<j_p^{!}<j_{p+1}^{!}=j_i$, for $p\in \mathbb{N}$.

Let $I_i^1=(j_{i-1},j_p]$, $I_i^2=(j_p,j_i]$, $h_i^1=j_p-j_{i-1}$, $h_i^2=j_i-j_p$. Since $I_i^1\subset I_i$ and $I_i^2\subset I_i$, both $h_i^1$ and $h_i^2$ tend to ∞ as $i\to \mathrm{\infty }$. We have

$$\begin{array}{c}\frac{1}{{\mu }_i^{\beta }}(h_i^{-1}\sum _{j\in J_i}|x_j-M|)\hfill \\ \le \frac{1}{{\mu }_i^{\beta }}(\left(h_i^{-1}{h}_i^1\right){\left(h_i^1\right)}^{-1}\sum _{j\in I_i^1}|x_j-M|+\left(h_i^{-1}{h}_i^2\right){\left(h_i^2\right)}^{-1}\sum _{j\in I_i^2}|x_j-M|)\hfill \\ \le \left(\frac{{\mu }_i-{\lambda }_i}{{\mu }_i^{\beta }}\right)\left(h_i^{-1}{h}_i^1\right){\left(h_i^1\right)}^{-1}L+\frac{1}{{\mu }_i^{\beta }}(\left(h_i^{-1}{h}_i^2\right){\left(h_i^2\right)}^{-1}\sum _{j\in I_i^2}|x_j-M|)\hfill \\ \le \left(\frac{{\mu }_i-{\lambda }_i^{\beta }}{{\lambda }_i^{\beta }}\right)\left(h_i^{-1}{h}_i^1\right){\left(h_i^1\right)}^{-1}L+\frac{1}{{\mu }_i^{\beta }}(\left(h_i^{-1}{h}_i^2\right){\left(h_i^2\right)}^{-1}\sum _{j\in I_i^2}|x_j-M|)\hfill \\ \le (\frac{{\mu }_i}{{\lambda }_i^{\beta }}-1)\left(h_i^{-1}{h}_i^1\right){\left(h_i^1\right)}^{-1}L+\frac{1}{{\mu }_i^{\beta }}(\left(h_i^{-1}{h}_i^2\right){\left(h_i^2\right)}^{-1}\sum _{j\in I_i^2,|x_j-M|\ge \epsilon }|x_j-M|)\hfill \\ +\frac{1}{{\mu }_i^{\beta }}(\left(h_i^{-1}{h}_i^2\right){\left(h_i^2\right)}^{-1}\sum _{j\in I_i^2,|x_j-M|<\epsilon }|x_j-M|)\hfill \\ \le (\frac{{\mu }_i}{{\lambda }_i^{\beta }}-1)\left(h_i^{-1}{h}_i^1\right){\left(h_i^1\right)}^{-1}L+\frac{L}{{\lambda }_i^{\alpha }}\left|\{j\in I_i:\left(h_i^{-1}{h}_i^2\right){\left(h_i^2\right)}^{-1}|x_j-M|\ge \epsilon \}\right|\hfill \\ +\epsilon \left(h_i^{-1}{h}_i^2\right){\left(h_i^2\right)}^{-1},\mathrm{\forall }i\in \mathbb{N}\hfill \\ =\frac{\delta }{2}\left(h_i^{-1}{h}_i^1\right){\left(h_i^1\right)}^{-1}L+\frac{L}{{\lambda }_i^{\alpha }}\left|\{j\in I_i:\left(h_i^{-1}{h}_i^2\right){\left(h_i^2\right)}^{-1}|x_j-M|\ge \epsilon \}\right|+\epsilon \left(h_i^{-1}{h}_i^2\right){\left(h_i^2\right)}^{-1}.\hfill \end{array}$$

Since $x\in w_{I_{\mu }}^{\beta }(\mathcal{M},\theta !)$, we have $0<h_i^{-1}{h}_i^1\le 1$ and $0<h_i^{-1}{h}_i^2\le 1$.

Hence, for $\xi >0$,

$$\begin{array}{rl}\{i\in \mathbb{N}:\frac{1}{{\mu }_i^{\beta }}(\frac{1}{h_i}\sum _{j\in J_i}|x_j-M|\ge \gamma )\ge \xi \}& \subset \{i\in \mathbb{N}:\frac{L}{{\lambda }_i^{\alpha }}|\{j\in I_i:\frac{1}{h_i^2}|x_j-M|\ge \gamma \}|\ge \xi \}\\ & \cup \{1,2,3,\dots ,s\}.\end{array}$$

Since $(x_j)$ is lacunary $I_{\lambda }$-statistically convergent sequence of order α defined by $\mathcal{M}$ and since I is admissible, by using the continuity of $\mathcal{M}$, we can say that

$$S_{I_{\lambda }}^{\alpha }(\mathcal{M},\theta )\subseteq w_{I_{\mu }}^{\beta }(\mathcal{M},\theta !).$$

 □

Corollary 3.1

Let $\lambda \le {\mu }_i$ for all $i\in \mathbb{N}$ and $0<\alpha \le \beta \le 1$. Let $lim{inf}_{i\to \mathrm{\infty }}\frac{{\lambda }_i^{\alpha }}{{\mu }_i^{\beta }}>0$, θ! be the refinement of θ. Also let $\mathcal{M}=(M_i)$ be a Musielak-Orlicz function where $(M_i)$ is pointwise convergent. Then $w_{I_{\mu }}^{\beta }(\mathcal{M},\theta !)\subset S_{I_{\lambda }}^{\alpha }(\mathcal{M},\theta )$ iff ${lim}_i{M}_i(\frac{\gamma }{{\rho }^{(i)}})>0$, for some $\gamma >0$, ${\rho }^{(i)}>0$.

Corollary 3.2

Let $\mathcal{M}=(M_i)$ be a Musielak-Orlicz function and ${lim}_{i\to \mathrm{\infty }}\frac{{\mu }_i}{{\lambda }_i^{\beta }}=1$, for fixed numbers α and β such that $0<\alpha \le \beta \le 1$. Then $S_{I_{\lambda }}^{\alpha }(\mathcal{M},\theta )\subset w_{I_{\mu }}^{\beta }(\mathcal{M},\theta )$ iff ${sup}_{\nu }{sup}_i(\frac{\nu }{{\rho }^{(i)}})$.

Lacunary $I_{\lambda }$-statistical convergence in probabilistic normed spaces

Let X be a real linear space and $\nu :X\to D$, where D is the set of all distribution functions $g:\mathbb{R}\to {\mathbb{R}}_0^{+}$ such that it is non-decreasing and left-continuous with ${inf}_{t\in \mathbb{R}}g(t)=0$ and ${sup}_{t\in \mathbb{R}}g(t)=1$. The probabilistic norm or ν-norm is a t-norm [21] satisfying the following conditions:

1. ${\nu }_p(0)=0$,

2. ${\nu }_p(t)=1$ for all $t>0$ iff $p=0$,

3. ${\nu }_{\alpha p}(t)={\nu }_p(\frac{t}{|\alpha |})$ for all $\alpha \in \mathbb{R}\mathrm{\setminus }\{0\}$ and for all $t>0$,

4. ${\nu }_{p+q}(s+t)\ge \tau ({\nu }_p(s),{\nu }_q(t))$ for all $p,q\in X$ and $s,t\in {\mathbb{R}}_0^{+}$;

$(X,\nu ,\tau )$ is named a probabilistic normed space, in short PNS.

A sequence $x=(x_i)$ is I-convergent to $M\in X$ in $(X,\nu ,\tau )$ for each $\xi >0$ and $t>0$, $\{i\in \mathbb{N}:{\nu }_{x_i-M}(t)\le 1-\xi \}\in I$ (here I is a non-trivial ideal of $\mathbb{N}$) [19].

We define a sequence $x=(x_i)$ to be lacunary $I_{\lambda }$-statistical convergent to M in $(X,\nu ,\tau )$ defined by $\mathcal{M}$, if, for each $\nu >0$, $M>0$, $\mu >0$, $\xi >0$ and $t>0$,

$$\{i\in \mathbb{N}:\frac{1}{{\lambda }_i}|\{j\le i:\frac{1}{h_i}\sum _{j\in I_i}{M}_j\left(\frac{{\nu }_{x_j-M}(t)}{{\rho }^{(j)}}\right)\le 1-\mu \}|\le 1-\xi \}\in I.$$

We write it as $I_{\lambda }^{\nu }(\theta )limx=\psi$.

Example: Let $(\mathbb{R},\nu ,\tau )$ be a PNS with the probabilistic norm ${\nu }_p(t)=\frac{t}{t+|p|}$ (for all $p\in \mathbb{R}$ and every $t>0$) and $\tau (a,b)=ab$. Also, let I be a non-trivial admissible ideal such that $I=\{B\subset \mathbb{N}:\delta (B)=0\}$. Define a sequence x as follows:

$$x_i=\{\begin{array}{cc}\frac{1}{i}\hfill & \text{if }i=k^2\text{, }i\in \mathbb{N};\hfill \\ 0\hfill & \text{otherwise}.\hfill \end{array}$$

Then we have, for each $\nu >0$, $M>0$, $\mu >0$, $\xi >0$ and $t>0$, $\delta (K)=0$, where

$$K=\{i\in \mathbb{N}:\frac{1}{{\lambda }_i}|\{j\le i:\frac{1}{h_i}\sum _{j\in I_i}{M}_j\left(\frac{{\nu }_{x_j-M}(t)}{{\rho }^{(j)}}\right)\le 1-\mu \}|\le 1-\xi \},$$

which implies $K\in I$ and $I_{\lambda }^{\nu }(\theta )-lim=0$.

Theorem 4.1

Let $(X,\nu ,\tau )$ be a PNS. If $x=(x_i)$ is lacunary $I_{\lambda }^{\nu }$-statistical convergent, then it has a unique limit.

Proof

Suppose $x=(x_i)$ to be lacunary $I_{\lambda }^{\nu }$-statistical convergent in X which has two limits, $M_1$ and $M_2$.

For $\beta >0$ and $t>0$, let us choose $\xi >0$ such that $\tau ((1-\xi ),(1-\xi ))\ge 1-\beta$.

Take the following sets:

$$\begin{array}{c}{K}_1(\xi ,t)=\{i\in \mathbb{N}:\frac{1}{{\lambda }_i}|\{j\le i:\frac{1}{h_i}\sum _{j\in I_i}{M}_j\left(\frac{{\nu }_{x_j-M_1}(t)}{{\rho }^{(j)}}\right)\le 1-\mu \}|\le 1-\xi \},\hfill \\ K_2(\xi ,t)=\{i\in \mathbb{N}:\frac{1}{{\lambda }_i}|\{j\le i:\frac{1}{h_i}\sum _{j\in I_i}{M}_j\left(\frac{{\nu }_{x_j-M_2}(t)}{{\rho }^{(j)}}\right)\le 1-\mu \}|\le 1-\xi \}.\hfill \end{array}$$

Since $x=(x_i)$ is lacunary $I_{\lambda }^{\nu }$-statistical convergent to $M_1$, we have $K_1(\xi ,t)\in I$. Similarly, $K_2(\xi ,t)\in I$.

Now, let $K(\xi ,t)=K_1(\xi ,t)\cup K_2(\xi ,t)\in I$. We see that $K(\xi ,t)$ belongs to I, from which it is clear that $K^C(\xi ,t)$ is non-empty set in $F(I)$, where $F(I)$ is the filter associated with the ideal I [9].

If $i\in K^C(\xi ,t)$, then we have $i\in K_1^C(\xi ,t)\cap K_2^C(\xi ,t)$ and so

$${\nu }_{M_1-M_2}(t)\ge \tau ({\nu }_{x_i-M_1}\left(\frac{t}{2}\right),{\nu }_{x_i-M_2}\left(\frac{t}{2}\right))>\tau ((1-\xi ),(1-\xi )).$$

Since $\tau ((1-\xi ),(1-\xi ))\ge 1-\beta$, it follows that ${\nu }_{M_1-M_2}(t)>1-\beta$.

For arbitrary $\beta >0$, we get ${\nu }_{M_1-M_2}(t)=1$ for all $t>0$, which proves $M_1=M_2$. □

Theorem 4.2

Let $(X,\nu ,\tau )$ be a PNS. If x is lacunary $I^{\nu }$-statistical convergent, then it is lacunary $I_{\lambda }^{\nu }$-statistical convergent if ${lim}_i\frac{{\lambda }_i}{i}>0$.

Proof

For given $\mu >0$, $\xi >0$, and $t>0$,

$$\{j\le i:\frac{1}{h_i}\sum _{j\in I_i}{M}_j\left(\frac{{\nu }_{x_j-M}(t)}{{\rho }^{(j)}}\right)\le 1-\mu \}\supset \{j\in I_i:\frac{1}{h_i}\sum _{j\in I_i}{M}_j\left(\frac{{\nu }_{x_j-M}(t)}{{\rho }^{(j)}}\right)\le 1-\mu \}.$$

Therefore,

$$\begin{array}{c}\frac{1}{i}\{j\le i:\frac{1}{h_i}\sum _{j\in I_i}{M}_j\left(\frac{{\nu }_{x_j-M}(t)}{{\rho }^{(j)}}\right)\le 1-\mu \}\hfill \\ \ge \frac{1}{i}\{j\in I_i:\frac{1}{h_i}\sum _{j\in I_i}{M}_j\left(\frac{{\nu }_{x_j-M}(t)}{{\rho }^{(j)}}\right)\le 1-\mu \}\hfill \\ \ge \frac{1}{{\lambda }_i}.\frac{{\lambda }_i}{i}\{j\in I_i:\frac{1}{h_i}\sum _{j\in I_i}{M}_j\left(\frac{{\nu }_{x_j-M}(t)}{{\rho }^{(j)}}\right)\le 1-\mu \},\hfill \\ \{i\in \mathbb{N}:\frac{1}{i}\{j\le i:\frac{1}{h_i}\sum _{j\in I_i}{M}_j\left(\frac{{\nu }_{x_j-M}(t)}{{\rho }^{(j)}}\right)\le 1-\mu \}\le 1-\xi \}\hfill \\ \ge \frac{{\lambda }_i}{i}\{i\in \mathbb{N}:\frac{1}{{\lambda }_i}\{j\in I_i:\frac{1}{h_i}\sum _{j\in I_i}{M}_j\left(\frac{{\nu }_{x_j-M}(t)}{{\rho }^{(j)}}\right)\le 1-\mu \}\le 1-\xi \}.\hfill \end{array}$$

Since ${lim}_i\frac{{\lambda }_i}{i}>0$ and taking the limit $i\to \mathrm{\infty }$, we get $I_{\lambda }^{\nu }(\theta )-limx=M$. □

We define $x=(x_i)$ to be lacunary λ-statistically convergent to M with respect to ν as

$$\delta \left(\{i\in \mathbb{N}:\frac{1}{{\lambda }_i}|\{j\le i:\frac{1}{h_i}\sum _{j\in I_i}{M}_j\left(\frac{{\nu }_{x_j-M}(t)}{{\rho }^{(j)}}\right)\le 1-\mu \}|\le 1-r\}\right)=0.$$

Theorem 4.3

Let $(X,\nu ,\tau )$ be a PNS.

1. If x is lacunary λ-statistically convergent to M, then it is also lacunary $I_{\lambda }^{\nu }$-statistically convergent to M.

2. If $I_{\lambda }^{\nu }(\theta )-limx=M_1$, $I_{\lambda }^{\nu }(\theta )-limy=M_2$, then $I_{\lambda }^{\nu }(\theta )-lim(x_k+y_k)=(M_1+M_2)$.

3. If $I_{\lambda }^{\nu }(\theta )-limx=M$,then $I_{\lambda }^{\nu }(\theta )-lim\alpha x=\alpha M$.

Theorem 4.4

Let $(X,\nu ,\tau )$ be a PNS. If x is lacunary λ-statistical convergent to M, then $I_{\lambda }^{\nu }(\theta )-limx=M$.

Proof

Let $x=(x_i)$ be lacunary λ-statistically convergent to M, then, for every $t>0$, $\xi >0$ and $\mu >0$, there exists $i_0\in \mathbb{N}$ such that

$$\delta \left(\{i\in \mathbb{N}:\frac{1}{{\lambda }_i}\{j\le i:\frac{1}{h_i}\sum _{j\in I_i}{M}_j\left(\frac{{\nu }_{x_j-\psi }(t)}{{\rho }^{(j)}}\right)\le 1-\mu \}\le 1-\xi \}\right)=0,$$

for all $i\ge i_0$. Therefore the set

$$B=\{i\in \mathbb{N}:\{j\le i:\frac{1}{h_i}\sum _{j\in I_i}{M}_j\left(\frac{{\nu }_{x_j-\psi }(t)}{{\rho }^{(j)}}\right)\le 1-\mu \}\le 1-\xi \}\subseteq \{1,2,3,\dots i_0-1\}.$$

Since I is admissible, we have $B\in I$. Hence $I_{\lambda }^{\nu }(\theta )-limx=\psi$. □

Theorem 4.5

Let $(X,\nu ,\tau )$ be a PNS. If x is lacunary λ-statistical convergent, then it has a unique limit.

Theorem 4.6

Let $(X,\nu ,\tau )$ be a PNS. If x is lacunary λ-statistically convergent, then there exists a subsequence $(x_{m_k})$ of x such that it is also lacunary λ-statistically convergent to M.