Palestine Journal of Mathematics Vol. 5(1) (2016) , 50–58 © Palestine Polytechnic University-PPU 2016 On Generalized Statistical Convergence Maya Altınok, Zehra Kurtdi¸ si and Mehmet Küçükaslan Communicated by Ayman Badawi MSC 2010 Classifications: 40A05, 40C05, 10H25. Keywords and phrases: statistical convergence, summability of sequences, strongly summability, statistical Nörlund convergence, weighted statistical convergence. Abstract In this paper, statistical convergence is generalized by using regular Nörlund mean N(p) where p =(p n ) is a positive sequence of natural numbers. It is called statistical Nörlund convergence and denoted by the symbol st-N(p). Besides convergence properties of st-N(p), some inclusion results have been given between st-N(p) convergence and strongly N(p) and statistical convergence. Also, st-N(p) and st-N(q) convergences are compared under some certain restrictions. 1 Introduction and Backround Statistical convergence of real (or complex) valued sequences was first introduced by Fast H. [6] and Steinhaus I. J. [18] independently in the 1951 as a generalization of ordinary convergence. This subject has been applied various field of mathematics by many authors such as Erdös P.- Terenbaum G. [5], Freeedman A. R.- Sember J. J. - Raphael M. [7], etc. Besides, Connor J.[3, 4], Fridy J. A. [8], Fridy J. A.- Orhan C. [9], Salat T. [16], Schenberg I. J. [17]. Statistical convergence is closely related to the natural density of the subset K of natural numbers N ( see more in [2] ). For n ∈ N, let K(n) := {k | k ≤ n, k ∈ K} for K ⊂ N. Then, the natural density (or asymptotic density) of K ⊆ N is denoted by δ(K), and δ(K) := lim n→∞ 1 n n  k=1 χ K(n) (k) (1.1) if this limit exists. In (1.1) the symbol χ K(n) (.) denotes the characteristic function of the set K(n). A real (or complex) sequence x =(x n ) is said to be statistical convergent to l ∈ R (∈ C), if the set K(ε) := {k | k ≤ n, |x k − l|≥ ε} has natural density zero for every ε> 0, i. e., δ(K(ε)) = 0. This limit is denoted by x n → l(st). Throughout this paper, let p =(p n ) be a sequence of nonnegative natural numbers with p 0 = 0 and p n > 0 for all n ∈ N and P n := ∑ n k=0 p k . The Nörlund mean of the sequence x =(x n ) is defined by t n := 1 Pn ∑ n k=1 p n-k+1 x k . The sequence x =(x n ) is said to be N (p) convergent to l ∈ R if the sequence (t n ) n∈N convergent to l ∈ R, and strongly N(p) convergence to l if lim n→∞ 1 P n n  k=1 p n-k+1 |x k − l| = 0. and it is denoted by x n → l (N(p)). Definition 1.1. The sequence x =(x n ) is said to be statistically Nörlund convergent to l if lim n→∞ 1 P n n  k=1 p n-k+1 χ K(ε) (k)= 0, (1.2) holds. It is denoted by x n → l (st-N(p)). The case p n = 1 in Definition 1.1 is coincide with usual statistical convergence [8]. This kind of generalization of statistical convergence has been given by Fredman and Sember in [7] by using any regular matrix summability method.