Some results concerning ideal and classical uniform integrability and mean convergence
Tóm tắt
In this article, the concept of
$${\mathcal {J}}$$
-uniform integrability of a sequence of random variables
$$\left\{ X_{k}\right\} $$
with respect to
$$\left\{ a_{nk} \right\} $$
is introduced where
$${\mathcal {J}}$$
is a non-trivial ideal of subsets of the set of positive integers and
$$\left\{ a_{nk} \right\} $$
is an array of real numbers. We show that this concept is weaker than the concept of
$$\left\{ X_{k} \right\} $$
being uniformly integrable with respect to
$$\left\{ a_{nk} \right\} $$
and is more general than the concept of B-statistical uniform integrability with respect to
$$\left\{ a_{nk} \right\} $$
. We give two characterizations of
$${\mathcal {J}}$$
-uniform integrability with respect to
$$\left\{ a_{nk} \right\} $$
. One of them is a de La Vallée Poussin type characterization. For a sequence of pairwise independent random variables
$$\left\{ X_{k} \right\} $$
which is
$${\mathcal {J}}$$
-uniformly integrable with respect to
$$\left\{ a_{nk} \right\} $$
, a law of large numbers with mean ideal convergence is proved. We also obtain a version without the pairwise independence assumption by strengthening other conditions. Supplements to the classical Mean Convergence Criterion are also established.
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