On Asymptotically Double Lacunary Statistically Equivalent Sequences

Abstract

This work presents the following definition which is a natural combination of the definition for asymptotically equivalent, double statistically limited and double lacunary sequences. Let theta(r s) = {(k(r), l(s))} be a double lacunary sequence; the two nonnegative sequences x = (x(k l)) and y = (y(k l)) are said to be asymptotically double lacunary statistically equivalent of multiple L provided that for every epsilon > 0

P - lim(r,s) 1/h(r,s) |{(k, l) is an element of l(r,s) : |x(k,l)/y(k l) - L| >= epsilon}| = 0

(denoted by x similar to(S theta r,sL) y) and simply asymptotically double lacunary statistically equivalent if L = 1. (C) 2009 Elsevier Ltd. All rights reserved.