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.