On the strong laws of large numbers for double arrays of random variables in convex combination spaces

Abstract

Let X be a convex combination space as defined by Terán and Molchanov [13]. By using their definition of mathematical expectation of an X-valued random variable, we state several new variants of strong laws of large numbers for double arrays of integrable X-valued random variables under various assumptions. Some related results in the literature are extended.

Strong laws of large numbers for sequences of blockwise and pairwise m-dependent random variables in metris spaces

“…random variables, which extended Theorem 1 of Etemadi [3]. Some more limit theorems for random variables taking values in a convex combination space can be found in Quang and Thuan [6], Thuan et al [9].…”

“…Let {X, X n : n 1} be a sequence of pairwise m-dependent and identically distributed X-valued random variables. Then condition (5) implies (6).…”

Approach for a metric space with a convex combination operation and applications

“…Lemma 3.3 in [18] established an inequality in CC space and it is a useful tool to obtain many limit theorems (see [18,23]). Now by applying Theorem 3.3, this lemma may be proved more easily as follows:…”

“…Furthermore, the authors also established the Etemadi strong law of large numbers (SLLN) for normalized sums of pairwise independent, identically distributed (i.i.d.) random elements in this kind of space ( [21], Theorem 5.1), other applications can be found in [18,22,23].…”

Baum-Katz’s Type Theorems for Pairwise Independent Random Elements in Certain Metric Spaces

“…random elements [12,Theorem 5.1], which extended [4, Theorem 1] of Etemadi. Since then, some limit theorems for random elements taking values in convex combination space were considered and extended (see [9,11,12,14]). On the other hand, as shown recently in [13], it is fairly remarkable that although these spaces are not linear in general, they always contains a subspace which can be isometrically embedded into a Banach space and this embedding preserves the convex combination operation.…”