A few remarks on divergent sequences: Rates of divergence II

Abstract

We continue (and, in a sense, close) our investigation begun in the paper [10] concerning quotient speed of divergence of sequences of positive real numbers and its relations with selection principles and games. By the way, we show a theorem of the generalized Galambos-Bojanić-Seneta type.

Double Sequences and Selections

“…In this paper we begin with a quite different investigation of double sequences related to selection principles and games corresponded to them : for a given sequence X n : n ∈ N of double sequences that belong to one class A we select from each X n a subset Y n by a prescribed procedure so that Y n 's may be arranged to a new double sequence Y which belongs to another class B not necessarily distinct from A of double sequences. For selection principles theory see 14, 15 , and for selection properties of some classes of simple sequences see [16][17][18][19] . Moreover, our investigation suggests also introduction of new selection principles: instead of a sequence of double sequences from a class A we start with a double sequence of double sequences from A see Definitions 2.1 and 2.7 .…”

A note on convergence in measure and selection principles

“…In a number of papers by the authors that appeared recently in the literature it was demonstrated that some classes A and B of sequences of positive real numbers have certain nice selection properties ( [3][4][5][6][7]).…”

The weak and the strong equivalence relation and the asymptotic inversion

“…The class P I * is a subclass of the class P I (see e.g. [5]) More information about these classes can be found in [7] and [12].…”