\(\mathcal{I}_\theta\)-STATISTICALLY CONVERGENT SEQUENCES IN TOPOLOGICAL GROUPS

Abstract

Recently, Das, Savaş and Ghosal [4], defined the lacunary statistical analogue for the sequence \(x=(x_k)\) as follows: A real sequence \(x=(x_k)\) is said to be \(\mathcal{I}\)-lacunary statistically convergent to \(L\) or \(S_\theta^{\mathcal{I}}\)-convergent to \(L\) if for any \(\epsilon>0\) and \(\delta>0\)
\[
\left\{r\in\mathbb{N}:\frac{1}{h_r}\left|\left\{k\in I_r:|x_k-L|\geq\epsilon\right\}\right|\geq\delta\right\}\in\mathcal{I}.
\]
In this case write \(S_\theta^{\mathcal{I}}-\lim x=L\) or \(x_k\to L(S_\theta^{\mathcal{I}})\). In this paper, we introduce and study \(\mathcal{I}\)-lacunary statistical convergence for sequence in topological groups and we shall also present some inclusion theorems.