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 [2–8] and many more.

Similarly, I-convergence is also an extended notion of statistical convergence ([9]) of real sequences. A family of sets \(I \subseteq2^{A}\) (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 [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 \(\phi_{j}\).

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.