Open Access

Some new lacunary statistical convergence with ideals

Journal of Inequalities and Applications20172017:15

https://doi.org/10.1186/s13660-016-1284-9

Received: 29 August 2016

Accepted: 23 December 2016

Published: 10 January 2017

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.

Keywords

Musielak-Orlicz function ideal convergence lacunary sequences probabilistic normed space

1 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 [28] and many more.

Similarly, I-convergence is also an extended notion of statistical convergence ([9]) of real sequences. A family of sets \(I \subseteq2^{A}\) (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 \(h_{j}=i_{j}-i_{j-1} \rightarrow\infty\) as \(j \rightarrow\infty\). As regards ideal convergence and lacunary ideal convergence, one may refer to [1019] 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 \(\phi_{j}\).

2 Preliminary concepts

A sequence \((x_{i})\) of real numbers is statistically convergent to M if, for arbitrary \(\xi>0\), the set \(K(\xi)=\{i \in\mathbb{N}: \vert x_{i} -M \vert\geq\xi\}\) has natural density zero, i.e.,
$$\lim_{i} \frac{1}{i} \sum_{j=1}^{i} \chi_{K(\xi)}(j)=0, $$
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 \(\xi>0\),
$$\bigl\{ i \in\mathbb{N}: \vert x_{i} -M \vert \geq\xi\bigr\} \in I. $$
For any lacunary sequence \(\theta= (i_{j})\), the space \(N_{\theta}\) is defined as (Freedman et al. [5])
$$N_{\theta}= \biggl\{ (x_{i}): \lim_{j \rightarrow\infty} i_{j}^{-1} \sum_{i \in K_{j}} \vert x_{i} - M \vert =0, \mbox{ for some } M \biggr\} . $$
The concept of a Musielak-Orlicz function is defined as \(\mathscr {M}=(M_{j})\). The sequence \(\mathscr{N}=(N_{i})\) is defined by
$$N_{i}(a)=\sup\bigl\{ \vert a \vert b -M_{j}(b): b \geq0 \bigr\} ,\quad i=1,2,\ldots, $$
which is named the complementary function of a Musielak-Orlicz function \(\mathscr{M}\) (see [20]) (throughout the paper \(\mathscr{M}\) 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 \(\xi>0\),
$$\biggl\{ i \in\mathbb{N}: \frac{1}{\lambda_{i}^{\alpha}}\biggl\vert \biggl\{ j \leq i: \frac{1}{h_{i}} \sum_{j \in I_{i}} M_{j} \biggl( \frac{\vert x_{j}-M \vert }{\rho^{(j)}} \biggr) \geq\gamma \biggr\} \biggr\vert \geq\xi \biggr\} \in I. $$

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

Some particular cases:
  1. 1.

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

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

     
  2. 2.

    If \(\lambda_{i}=i\), for all \(i \in\mathbb{N}\), then \(S_{I_{\lambda}}^{\alpha}(\mathscr{M}, \theta)\) will be reduced to \(S_{I}^{\alpha}(\mathscr {M}, \theta)\).

     
  3. 3.

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

     
  4. 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. \((x_{j}) \in S_{I_{\lambda}}(\mathscr{M})\).

     
  5. 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 \(\mathscr{M}\) 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.

3 Main results

Theorem 3.1

Let \(\lambda=(\lambda_{i})\) and \(\mu=(\mu_{i})\) be two sequences in Λ such that \(\lambda_{i} \leq\mu_{i}\) for all \(i \in \mathbb{N}\) and \(0<\alpha\leq\beta\leq1\) for fixed reals α and β. If \(\lim\inf_{i \rightarrow\infty} \frac{\lambda_{i}^{\alpha}}{\mu_{i}^{\beta}} >0\), then \(S_{I_{\mu}}^{\beta}(\mathscr{M}, \theta) \subseteq S_{I_{\lambda}}^{\alpha}(\mathscr{M}, \theta)\).

Proof

Suppose that \(\lambda_{i} \leq\mu_{i}\) for all \(i \in\mathbb {N}\) and \(\lim\inf_{i \rightarrow\infty} \frac{\lambda_{i}^{\alpha}}{\mu _{i}^{\beta}} >0\). Since \(I_{i} \subset J_{i}\), where \(J_{i}=[i-\mu_{i}+1, i]\), so for \(\gamma>0\), we can write
$$\bigl\{ j \in J_{i}: \vert x_{j} -M \vert \geq\gamma \bigr\} \supset\bigl\{ j \in I_{i}:\vert x_{j} -M \vert \geq\gamma\bigr\} , $$
which implies
$$\frac{1}{\mu_{i}^{\beta}} \bigl\vert \bigl\{ j \in J_{i}: \vert x_{j} - M \vert \geq\gamma \bigr\} \bigr\vert \geq\frac{\lambda_{i}^{\alpha}}{\mu _{i}^{\beta}}. \frac{1}{\lambda_{i}^{\alpha}} \bigl\vert \bigl\{ j \in I_{i}:\vert x_{j} - M \vert \geq\gamma \bigr\} \bigr\vert , $$
for all \(i \in\mathbb{N}\).
Assume that \(\lim\inf_{i \rightarrow\infty} \frac{\lambda_{i}^{\alpha}}{\mu_{i}^{\beta}}=a\), 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 \(\xi>0\),
$$\begin{aligned} \biggl\{ i \in\mathbb{N}:\frac{1}{\lambda_{i}^{\beta}} \bigl\vert \bigl\{ j \in J_{i}: \vert x_{j} -M \vert \geq\gamma \bigr\} \bigr\vert \geq\xi \biggr\} \subset& \biggl\{ i \in\mathbb{N}: \frac{1}{\mu_{i}^{\alpha}} \bigl\vert \bigl\{ j \in I_{i} : \vert x_{j} -M \vert \geq\gamma \bigr\} \bigr\vert \geq\frac{a}{2} \xi \biggr\} \\ &{} \cup \biggl\{ i \in\mathbb{N} : \frac{\lambda_{i}^{\alpha}}{\mu_{i}^{\beta}} < \frac{a}{2} \biggr\} . \end{aligned}$$

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

Theorem 3.2

If \(\lim_{i \rightarrow\infty} \frac{\mu_{i}}{\lambda _{i}^{\beta}}=1\), for \(\lambda=(\lambda_{i})\) and \(\mu=(\mu_{i})\) two sequences of Λ such that \(\lambda_{i} \leq\mu_{i}\), \(\forall i \in\mathbb{N}\) and \(0<\alpha\leq\beta\leq1\) for fixed α, β reals, then \(S_{I_{\lambda}}^{\alpha}(\mathscr{M}, \theta) \subseteq S_{I_{\mu}}^{\beta}(\mathscr{M}, \theta)\).

Proof

Let \((x_{j})\) be lacunary \(I_{\lambda}\)-statistically convergent to M of order α defined by \(\mathscr{M}\). Also assume that \(\lim_{i \rightarrow\infty} \frac{\mu_{i}}{\lambda_{i}^{\beta}} =1\). Choose \(m \in\mathbb{N}\) such that \(\vert \frac{\mu _{i}}{\lambda_{i}^{\beta}}-1 \vert < \frac{\xi}{2}\), \(\forall i\geq m\).

Since \(I_{i} \subset J_{i}\), for \(\gamma>0\), we may write
$$\begin{aligned} \frac{1}{\mu_{i}^{\beta}} \bigl\vert \bigl\{ j \in J_{i}: \vert x_{j} -M \vert \geq\gamma \bigr\} \bigr\vert =& \frac{1}{\mu_{i}^{\beta}} \bigl\vert \bigl\{ i- \mu_{i}+1 \leq j \leq i-\lambda_{i} : \vert x_{j} -M \vert \geq\gamma \bigr\} \bigr\vert \\ &{}+ \frac{1}{\mu_{i}^{\beta}} \bigl\vert \bigl\{ j \in I_{i} :\vert x_{j} -M\vert\geq\gamma \bigr\} \bigr\vert \\ \leq& \frac{\mu_{i}-\lambda_{i}}{\mu_{i}^{\beta}} + \frac{1}{\mu_{i}^{\beta}} \bigl\vert \bigl\{ j \in I_{i} : \vert x_{j} -M \vert\geq\gamma \bigr\} \bigr\vert \\ \leq& \frac{\mu_{i}-\lambda_{i}^{\beta}}{\lambda_{i}^{\beta}} + \frac{1}{\mu _{i}^{\beta}} \bigl\vert \bigl\{ j \in I_{i} : \vert x_{j} -M \vert\geq \gamma \bigr\} \bigr\vert \\ \leq& \biggl( \frac{\mu_{i}}{\lambda_{i}^{\beta}}-1 \biggr) + \frac{1}{\lambda _{i}^{\alpha}} \bigl\vert \bigl\{ j \in I_{i}: \vert x_{j} -M \vert\geq\gamma \bigr\} \bigr\vert \\ =& \frac{\xi}{2} + \frac{1}{\lambda_{i}^{\alpha}} \bigl\vert \bigl\{ j \in I_{i} : \vert x_{j} - M \vert \geq\gamma \bigr\} \bigr\vert . \end{aligned}$$
Hence,
$$\begin{aligned} \biggl\{ i \in\mathbb{N}:\frac{1}{\mu_{i}^{\beta}} \bigl\vert \bigl\{ j \leq i: \vert x_{j} -M \vert \geq\gamma \bigr\} \bigr\vert \geq\xi \biggr\} \subset& \biggl\{ i \in\mathbb{N}: \frac{1}{\lambda_{i}^{\alpha}} \bigl\vert \bigl\{ j \in I_{i} : \vert x_{j} -M \vert\geq\gamma \bigr\} \bigr\vert \geq \frac{\xi}{2} \biggr\} \\ &{}\cup \{ 1,2,3,\ldots,m \}. \end{aligned}$$
Since \((x_{j})\) is lacunary \(I_{\lambda}\)-statistically convergent sequence of order α defined by \(\mathscr{M}\) and since I is admissible, by using the continuity of \(\mathscr{M}\), it follows that the set on the right hand side with the lacunary sequence \(\theta=(h_{i})\) belongs to I and
$$S_{I_{\lambda}}^{\alpha}(\mathscr{M}, \theta)\subseteq S_{I_{\mu}}^{\beta}(\mathscr{M}, \theta). $$
 □
We define the lacunary \(I_{\lambda}\)-summable sequence of order α defined by \(\mathscr{M}\) as
$$w_{I_{\lambda}}^{\alpha}(\mathscr{M},\theta)= \biggl\{ i \in\mathbb{N}: \frac{1}{\lambda_{i}^{\alpha}} \biggl( j \leq i: \frac{1}{h_{i}} \sum _{j \in I_{i}} M_{j} \biggl( \frac{ \vert x_{j} - M \vert }{ \rho^{(j)}} \biggr) \geq \gamma \biggr) \biggr\} \in I. $$

Theorem 3.3

Given \(\lambda=(\lambda_{i})\), \(\mu=(\mu_{i}) \in\Lambda\). Suppose that \(\lambda_{i} \leq\mu_{i}\) for all \(i \in\mathbb{N}\), \(0 < \alpha\leq\beta\leq1\). Then:
  1. 1.

    If \(\lim\inf_{i \rightarrow\infty} \frac{\lambda_{i}^{\alpha}}{\mu _{i}^{\beta}} >0\), then \(w_{\mu}^{\beta}(\mathscr{M}, \theta) \subset w_{\lambda}^{\alpha}(\mathscr{M}, \theta)\).

     
  2. 2.

    If \(\lim_{i \rightarrow\infty} \frac{\mu_{i}}{\lambda_{i}^{\beta}} = 1\), then \(\ell_{\infty}\cap w_{\lambda}^{\alpha}(\mathscr{M},\theta) \subset w_{\mu}^{\beta}(\mathscr{M},\theta)\).

     

Theorem 3.4

Let \(\lambda_{i} \leq\mu_{i}\) for all \(i \in\mathbb{N}\), where \(\lambda, \mu\in\Lambda\). Then, if \(\lim\inf_{i \rightarrow \infty} \frac{\lambda_{i}^{\alpha}}{\mu_{i}^{\beta}} >0\), and if \((x_{j})\) is lacunary \(I_{\mu}\)-summable of order β defined by \(\mathscr{M}\), then it is lacunary \(I_{\lambda}\)-statistically convergent of order α defined by \(\mathscr{M}\). Here \(0 <\alpha\leq\beta\leq1\), for fixed reals α and β.

Proof

For any \(\gamma>0\), we have
$$\begin{aligned} \sum_{j \in J_{i}} \vert x_{j} -M \vert =& \sum_{j \in J_{i}, \vert x_{j} -M \vert\geq\varepsilon} \vert x_{j} -M \vert+ \sum _{j \in J_{i}, \vert x_{j} -M \vert< \varepsilon} \vert x_{j} -M \vert \\ \geq& \sum_{j \in I_{i}, \vert x_{j} - M \vert\geq\varepsilon} \vert x_{j} - M \vert+ \sum_{j \in I_{i}, \vert x_{j} -M \vert\geq\varepsilon} \vert x_{j} -M \vert \\ \geq& \sum_{j \in I_{i}, \vert x_{j} -M \vert\geq\varepsilon} \vert x_{j} -M \vert \\ \geq& \bigl\vert \bigl\{ j \in I_{i}: \vert x_{j} -M \vert\geq\gamma \bigr\} \bigr\vert . \gamma. \end{aligned}$$
Therefore,
$$\begin{aligned} \frac{1}{\mu_{i}^{\beta}} \sum_{j \in J_{i}} \vert x_{j} -M \vert \geq& \frac {1}{\mu_{i}^{\beta}} \bigl\vert \bigl\{ j \in I_{i} : \vert x_{j} - M \vert\geq\gamma \bigr\} \bigr\vert . \gamma \\ \geq&\frac{\lambda_{i}^{\alpha}}{\mu_{i}^{\beta}}. \frac{1}{\lambda_{i}^{\alpha}} \bigl\vert \bigl\{ j \in I_{i}: \vert x_{j} - M \vert\geq\gamma\bigr\} \bigr\vert . \gamma. \end{aligned}$$
If \(\lim\inf_{i \rightarrow\infty} \frac{\lambda_{i}^{\alpha}}{\mu_{i}^{\beta}} =a\), then \(\{ i \in\mathbb{N}: \frac{\lambda_{i}^{\alpha}}{\mu _{i}^{\beta}} < \frac{a}{2} \}\) is finite. So, for \(\delta>0\), we get
$$\begin{aligned}& \biggl\{ i \in\mathbb{N}: \frac{1}{\lambda_{i}^{\alpha}}\biggl\vert \biggl\{ j \leq i: \sum _{j \in J_{i}} \vert x_{j} - M \vert\geq\gamma \biggr\} \biggr\vert \geq\xi \biggr\} \\& \quad \subset \biggl\{ i \in\mathbb{N}: \frac {1}{\mu_{i}^{\beta}} \bigl\{ j \in I_{i} : \vert x_{j} - M \vert \geq\gamma \bigr\} \geq\frac{a}{2} \xi \biggr\} \\& \qquad {}\cup \biggl\{ i \in\mathbb{N} : \frac{\lambda_{i}^{\alpha}}{\mu_{i}^{\beta}} < \frac{a}{2} \biggr\} . \end{aligned}$$

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

Theorem 3.5

Let \(\lim_{i \rightarrow\infty} \frac{\mu_{i}}{\lambda _{i}^{\beta}}=1\), where \(0< \alpha\leq\beta\leq1\) for fixed reals α and β and \(\lambda_{i} \leq\mu_{i}\), for all \(i \in\mathbb {N}\), where \(\lambda, \mu\in\Lambda\). 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 \(\mathscr{M}\), then it is also a lacunary \(I_{\mu}\)-summable sequence of order β defined by \(\mathscr{M}\). i.e. \(S_{I_{\lambda}}^{\alpha}(\mathscr{M},\theta) \subseteq w_{I_{\mu}}^{\beta}(\mathscr{M}, \theta!)\).

Proof

Suppose that \((x_{j})\) is lacunary \(I_{\lambda}\)-statistically convergent sequence of order α defined by \(\mathscr{M}\).

Given that \(\lim_{i \rightarrow\infty} \frac{\mu_{i}}{\lambda_{i}^{\beta}}=1\), we can choose \(s \in\mathbb{N}\) such that \(\vert \frac{\mu _{i}}{\lambda_{i}^{\beta}}-1 \vert < \frac{\delta}{2}\), \(\forall i \geq s\).

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 \(i \rightarrow\infty\). We have
$$\begin{aligned}& \frac{1}{\mu_{i}^{\beta}} \biggl(h_{i}^{-1} \sum _{j \in J_{i}} \vert x_{j} -M \vert \biggr) \\& \quad \leq \frac{1}{\mu_{i}^{\beta}} \biggl(\bigl(h_{i}^{-1}h_{i}^{1} \bigr) \bigl(h_{i}^{1}\bigr)^{-1} \sum _{j \in I_{i}^{1}} \vert x_{j} -M \vert + \bigl(h_{i}^{-1}h_{i}^{2}\bigr) \bigl(h_{i}^{2}\bigr)^{-1} \sum _{j \in I_{i}^{2}} \vert x_{j} -M \vert \biggr) \\& \quad \leq \biggl(\frac{\mu_{i} - \lambda_{i}}{\mu_{i}^{\beta}} \biggr) \bigl(h_{i}^{-1}h_{i}^{1} \bigr) \bigl(h_{i}^{1}\bigr)^{-1}L+ \frac{1}{\mu_{i}^{\beta}} \biggl(\bigl(h_{i}^{-1}h_{i}^{2} \bigr) \bigl(h_{i}^{2}\bigr)^{-1} \sum _{j \in I_{i}^{2}} \vert x_{j} -M \vert \biggr) \\& \quad \leq \biggl(\frac{\mu_{i} - \lambda_{i}^{\beta}}{\lambda_{i}^{\beta}} \biggr) \bigl(h_{i}^{-1}h_{i}^{1} \bigr) \bigl(h_{i}^{1}\bigr)^{-1} L+ \frac{1}{\mu_{i}^{\beta}} \biggl(\bigl(h_{i}^{-1}h_{i}^{2} \bigr) \bigl(h_{i}^{2}\bigr)^{-1} \sum _{j \in I_{i}^{2}} \vert x_{j} -M \vert \biggr) \\& \quad \leq \biggl(\frac{\mu_{i} }{\lambda_{i}^{\beta}}-1 \biggr) \bigl(h_{i}^{-1}h_{i}^{1} \bigr) \bigl(h_{i}^{1}\bigr)^{-1} L+ \frac{1}{\mu_{i}^{\beta}} \biggl(\bigl(h_{i}^{-1}h_{i}^{2} \bigr) \bigl(h_{i}^{2}\bigr)^{-1} \sum _{j \in I_{i}^{2}, \vert x_{j} - M \vert \geq\varepsilon} \vert x_{j} -M \vert \biggr) \\& \qquad {} + \frac{1}{\mu_{i}^{\beta}} \biggl(\bigl(h_{i}^{-1}h_{i}^{2} \bigr) \bigl(h_{i}^{2}\bigr)^{-1} \sum _{j \in I_{i}^{2}, \vert x_{j} - M \vert < \varepsilon} \vert x_{j} -M \vert \biggr) \\& \quad \leq \biggl(\frac{\mu_{i}}{\lambda_{i}^{\beta}}-1 \biggr) \bigl(h_{i}^{-1}h_{i}^{1} \bigr) \bigl(h_{i}^{1}\bigr)^{-1} L+ \frac{L}{\lambda_{i}^{\alpha}} \bigl\vert \bigl\{ j \in I_{i}: \bigl(h_{i}^{-1}h_{i}^{2}\bigr) \bigl(h_{i}^{2}\bigr)^{-1} \vert x_{j}-M \vert \geq\varepsilon \bigr\} \bigr\vert \\& \qquad {} + \varepsilon\bigl(h_{i}^{-1}h_{i}^{2} \bigr) \bigl(h_{i}^{2}\bigr)^{-1},\quad \forall i \in \mathbb{N} \\& \quad = \frac{\delta}{2}\bigl(h_{i}^{-1}h_{i}^{1} \bigr) \bigl(h_{i}^{1}\bigr)^{-1} L+ \frac{L}{\lambda _{i}^{\alpha}} \bigl\vert \bigl\{ j \in I_{i}: \bigl(h_{i}^{-1}h_{i}^{2}\bigr) \bigl(h_{i}^{2}\bigr)^{-1} \vert x_{j}-M \vert \geq\varepsilon \bigr\} \bigr\vert + \varepsilon \bigl(h_{i}^{-1}h_{i}^{2} \bigr) \bigl(h_{i}^{2}\bigr)^{-1}. \end{aligned}$$

Since \(x \in w_{I_{\mu}}^{\beta}(\mathscr{M}, \theta!)\), we have \(0< h_{i}^{-1}h_{i}^{1}\leq1\) and \(0< h_{i}^{-1}h_{i}^{2} \leq1\).

Hence, for \(\xi>0\),
$$\begin{aligned} \biggl\{ i \in\mathbb{N}: \frac{1}{\mu_{i}^{\beta}} \biggl(\frac{1}{h_{i}} \sum _{j \in J_{i}} \vert x_{j}-M \vert \geq\gamma \biggr) \geq\xi \biggr\} &\subset \biggl\{ i \in\mathbb{N}: \frac{L}{\lambda_{i}^{\alpha}} \biggl\vert \biggl\{ j \in I_{i}: \frac{1}{h_{i}^{2}} \vert x_{j}-M \vert\geq\gamma \biggr\} \biggr\vert \geq\xi \biggr\} \\ &\quad {}\cup \{1,2,3,\ldots,s\}. \end{aligned} $$
Since \((x_{j})\) is lacunary \(I_{\lambda}\)-statistically convergent sequence of order α defined by \(\mathscr{M}\) and since I is admissible, by using the continuity of \(\mathscr{M}\), we can say that
$$S_{I_{\lambda}}^{\alpha}(\mathscr{M},\theta) \subseteq w_{I_{\mu}}^{\beta}(\mathscr{M}, \theta!). $$
 □

Corollary 3.1

Let \(\lambda \leq\mu_{i}\) for all \(i \in\mathbb{N}\) and \(0< \alpha\leq\beta\leq1\). Let \(\lim\inf_{i \rightarrow\infty } \frac{\lambda_{i}^{\alpha}}{\mu_{i}^{\beta}} >0\), θ! be the refinement of θ. Also let \(\mathscr{M}=(M_{i})\) be a Musielak-Orlicz function where \((M_{i})\) is pointwise convergent. Then \(w_{I_{\mu}}^{\beta}(\mathscr {M}, \theta!)\subset S_{I_{\lambda}}^{\alpha}(\mathscr{M}, \theta)\) iff \(\lim_{i} M_{i} (\frac{\gamma}{\rho^{(i)}} )>0\), for some \(\gamma >0\), \(\rho^{(i)}>0\).

Corollary 3.2

Let \(\mathscr{M}=(M_{i})\) be a Musielak-Orlicz function and \(\lim_{i \rightarrow\infty} \frac{\mu_{i}}{\lambda_{i}^{\beta}} =1\), for fixed numbers α and β such that \(0< \alpha\leq\beta \leq1\). Then \(S_{I_{\lambda}}^{\alpha}(\mathscr{M}, \theta) \subset w_{I_{\mu}}^{\beta}(\mathscr{M}, \theta)\) iff \(\sup_{\nu}\sup_{i} (\frac {\nu}{\rho^{(i)}} )\).

4 Lacunary \(I_{\lambda}\)-statistical convergence in probabilistic normed spaces

Let X be a real linear space and \(\nu: X \rightarrow D\), where D is the set of all distribution functions \(g:\mathbb{R} \rightarrow\mathbb {R}_{0}^{+}\) 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. 1.

    \(\nu_{p}(0)=0\),

     
  2. 2.

    \(\nu_{p}(t)=1\) for all \(t>0\) iff \(p=0\),

     
  3. 3.

    \(\nu_{\alpha p}(t)=\nu_{p} (\frac{t}{\vert \alpha \vert } )\) for all \(\alpha\in\mathbb{R}\backslash\{0 \}\) and for all \(t >0\),

     
  4. 4.

    \(\nu_{p+q}(s+t) \geq \tau(\nu_{p}(s),\nu_{q}(t))\) 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 \(t>0\), \(\{ i \in\mathbb{N}: \nu_{x_{i}- M }(t) \leq1-\xi\} \in I\) (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 \(\mathscr{M}\), if, for each \(\nu>0\), \(M>0\), \(\mu>0\), \(\xi>0\) and \(t>0\),
$$\biggl\{ i \in\mathbb{N}: \frac{1}{\lambda_{i}} \biggl\vert \biggl\{ j \leq i: \frac{1}{h_{i}} \sum_{j \in I_{i}} M_{j} \biggl( \frac{\nu_{x_{j}-M}(t)}{\rho ^{(j)}} \biggr) \leq1-\mu \biggr\} \biggr\vert \leq1-\xi \biggr\} \in I. $$

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

Example: Let \((\mathbb{R}, \nu, \tau)\) be a PNS with the probabilistic norm \(\nu_{p}(t)=\frac{t}{t+\vert p\vert }\) (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:
$$x_{i} = \textstyle\begin{cases} \frac{1}{i} & \mbox{if $i=k^{2}$, $i \in\mathbb{N}$};\\ 0 & \mbox{otherwise}. \end{cases} $$
Then we have, for each \(\nu>0\), \(M>0\), \(\mu>0\), \(\xi>0\) and \(t>0\), \(\delta (K)=0\), where
$$K= \biggl\{ i \in\mathbb{N}: \frac{1}{\lambda_{i}} \biggl\vert \biggl\{ j \leq i: \frac{1}{h_{i}} \sum_{j \in I_{i}} M_{j} \biggl( \frac{\nu_{x_{j}-M }(t)}{\rho^{(j)}} \biggr) \leq1-\mu \biggr\} \biggr\vert \leq1-\xi \biggr\} , $$
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 \(\tau((1-\xi), (1-\xi)) \geq1-\beta\).

Take the following sets:
$$\begin{aligned}& K_{1}(\xi,t)= \biggl\{ i \in\mathbb{N}: \frac{1}{\lambda_{i}}\biggl\vert \biggl\{ j \leq i: \frac{1}{h_{i}} \sum_{j \in I_{i}} M_{j} \biggl(\frac{\nu _{x_{j}-M_{1}}(t)}{\rho^{(j)}} \biggr) \leq1-\mu \biggr\} \biggr\vert \leq1-\xi \biggr\} , \\& K_{2}(\xi,t)= \biggl\{ i \in\mathbb{N}: \frac{1}{\lambda_{i}}\biggl\vert \biggl\{ j \leq i: \frac{1}{h_{i}} \sum_{j \in I_{i}} M_{j} \biggl(\frac{\nu _{x_{j}-M_{2}}(t)}{\rho^{(j)}} \biggr) \leq1-\mu \biggr\} \biggr\vert \leq1-\xi \biggr\} . \end{aligned}$$

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
$$\nu_{M_{1}-M_{2}}(t) \geq \tau \left(\nu_{x_{i}-M_{1}} \left( \frac {t}{2} \right) , \nu_{x_{i}-M_{2}} \left(\frac{t}{2} \right) \right) > \tau((1-\xi ) , (1-\xi)). $$

Since \(\tau((1-\xi), (1-\xi)) \geq1-\beta\), 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 \(t>0\),
$$\biggl\{ j \leq i: \frac{1}{h_{i}} \sum_{j \in I_{i}} M_{j} \biggl(\frac{\nu _{x_{j}-M}(t)}{\rho^{(j)}} \biggr) \leq1- \mu \biggr\} \supset \biggl\{ j \in I_{i} : \frac{1}{h_{i}} \sum _{j \in I_{i}} M_{j} \biggl(\frac{\nu_{x_{j}-M }(t)}{\rho^{(j)}} \biggr) \leq1- \mu \biggr\} . $$
Therefore,
$$\begin{aligned}& \frac{1}{i} \biggl\{ j \leq i: \frac{1}{h_{i}} \sum _{j \in I_{i}} M_{j} \biggl(\frac{\nu_{x_{j}-M}(t)}{\rho^{(j)}} \biggr) \leq1-\mu \biggr\} \\& \quad \geq \frac{1}{i} \biggl\{ j \in I_{i}: \frac{1}{h_{i}} \sum_{j \in I_{i}} M_{j} \biggl( \frac{\nu_{x_{j}-M}(t)}{\rho^{(j)}} \biggr) \leq1-\mu \biggr\} \\& \quad \geq \frac{1}{\lambda_{i}}.\frac{\lambda_{i}}{i} \biggl\{ j \in I_{i}: \frac {1}{h_{i}} \sum_{j \in I_{i}} M_{j} \biggl( \frac{\nu_{x_{j}-M}(t)}{\rho ^{(j)}} \biggr) \leq1-\mu \biggr\} , \\& \biggl\{ i \in\mathbb{N}: \frac{1}{i} \biggl\{ j \leq i: \frac{1}{h_{i}} \sum_{j \in I_{i}} M_{j} \biggl(\frac{\nu_{x_{j}-M}(t)}{\rho^{(j)}} \biggr) \leq1-\mu \biggr\} \leq1-\xi \biggr\} \\& \quad \geq\frac{\lambda_{i}}{i} \biggl\{ i \in\mathbb{N}: \frac{1}{\lambda_{i}} \biggl\{ j \in I_{i}: \frac{1}{h_{i}} \sum_{j \in I_{i}} M_{j} \biggl(\frac{\nu _{x_{j}-M}(t)}{\rho^{(j)}} \biggr) \leq1-\mu \biggr\} \leq1-\xi \biggr\} . \end{aligned}$$

Since \(\lim_{i} \frac{\lambda_{i}}{i}>0\) and taking the limit \(i \rightarrow\infty\), we get \(I_{\lambda}^{\nu}(\theta)- \lim x=M\). □

We define \(x=(x_{i})\) to be lacunary λ-statistically convergent to M with respect to ν as
$$\delta \biggl( \biggl\{ i \in\mathbb{N}: \frac{1}{\lambda_{i}} \biggl\vert \biggl\{ j \leq i: \frac{1}{h_{i}} \sum_{j \in I_{i}} M_{j} \biggl(\frac{\nu _{x_{j}-M}(t)}{\rho^{(j)}} \biggr) \leq1-\mu \biggr\} \biggr\vert \leq1-r \biggr\} \biggr)=0. $$

Theorem 4.3

Let \((X,\nu,\tau)\) be a PNS.
  1. 1.

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

     
  2. 2.

    If \(I_{\lambda}^{\nu}(\theta)- \lim x=M_{1}\), \(I_{\lambda}^{\nu}(\theta)- \lim y=M_{2}\), then \(I_{\lambda}^{\nu}(\theta)- \lim(x_{k}+y_{k})=(M_{1}+M_{2})\).

     
  3. 3.

    If \(I_{\lambda}^{\nu}(\theta)- \lim x=M\),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 \(I_{\lambda}^{\nu}(\theta )- \lim x=M\).

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
$$\delta \biggl( \biggl\{ i \in\mathbb{N}: \frac{1}{\lambda_{i}} \biggl\{ j \leq i : \frac{1}{h_{i}} \sum_{j \in I_{i}} M_{j} \biggl( \frac{\nu_{x_{j}-\psi }(t)}{\rho^{(j)}} \biggr) \leq1-\mu \biggr\} \leq1-\xi \biggr\} \biggr)=0, $$
for all \(i \geq i_{0}\). Therefore the set
$$B= \biggl\{ i \in\mathbb{N}: \biggl\{ j \leq i: \frac{1}{h_{i}} \sum _{j \in I_{i}} M_{j} \biggl(\frac{\nu_{x_{j}-\psi}(t)}{\rho^{(j)}} \biggr) \leq1-\mu \biggr\} \leq1-\xi \biggr\} \subseteq\{1,2,3,\ldots i_{0}-1 \}. $$

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

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.

Declarations

Acknowledgements

The authors would like to extend their sincere appreciation to the referees for very useful comments and remarks for the earlier version of the manuscript.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, University Putra Malaysia
(2)
Department of Mathematics, Indian Institute of Technology, Bombay

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