In this paper, we study the concepts of lacunary statistical convergent and lacunary statistical Cauchy sequences in probabilistic normed spaces and prove some basic properties.

The law of the iterated logarithm (LIL) first arose in the work of Khintchine [5] who sought to obtain the exact rate of convergence in Borel’s theorem on normal numbers. This result was generalized by Kolmogorov [6] to sums of independent random variables. Recall that an increasing sequence of positive numbers {nk} is said to satisfy the Hadamard gap condition if there exists a q > 1 such that...

In this article we introduce and study the notions of ∆(r)-lacunary strongly summable, ∆(r)-Cesàro strongly summable, ∆(r)-statistically convergent and ∆(r)-lacunary statistically convergent sequences of fuzzy numbers. Consequently we construct the spaces of these four types of difference sequences and investigate the relationship among these spaces. Further we show that the spaces of ∆(r)-lacu...

Abstract Let $F \subseteq [0,1]$ be a set that supports probability measure $\mu $ with the property |\widehat{\mu }(t)| \ll (\log |t|)^{-A}$ for some constant A> 0 $. $\mathcal{A}= (q_n)_{n\in{\mathbb{N}}} sequence of natural numbers. If $\mathcal{A}$ is lacunary and $A>2$, we establish quantitative inhomogeneous Khintchine-type theorem in which (1) points interest are restricted...

Recently in [22], Mursaleen introduced the concept of statistical convergence in random 2-normed spaces. In this paper, we define and study the notion of lacunary ∆-statistical convergence and lacunary ∆-statistical Cauchy sequences in random 2-normed spaces using lacunary density and prove some interesting theorems. Subjclass [2000] : Primary 40A05; Secondary 46A70, 40A99, 46A99.

In this paper‎, ‎following a very recent‎ ‎and new approach‎, ‎we further generalize recently introduced‎ ‎summability methods‎, ‎namely‎, ‎$I$-statistical convergence and‎ ‎$I$-lacunary statistical convergence (which extend the important‎ ‎summability methods‎, ‎statistical convergence and lacunary‎ ‎statistical convergence using ideals of $mathbb{N}$) and‎ ‎introduce the notions of $I$-statis...

Let ( a n ) = 1 ? $(a_{n})_{n=1}^{\infty }$ be lacunary sequence of positive real numbers. Rudnick and Technau showed that for almost all ? ? R $\alpha \in \mathbb {R}$ , the pair correlation $(\alpha a_{n})_{n=1}^{\infty mod is Poissonian. We show higher correlations hence nearest-neighbour spacing distribution are Poissonian as well, thereby extending result Zaharescu to real-valued sequences.

A primary example is to take an integer g ≥ 2 and set a(x) = g. As is true for any increasing sequence of integers, for almost every α the fractional parts αa(x) are uniformly distributed modulo 1. Moreover, for lacunary sequences, it has long been known that the fractional parts of αa(x) have strong randomness properties. For instance, the exponential sums 1 √ N ∑ x≤N cos(2παa(x)) have a Gauss...