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Since the appearance of H. Robbins article (1948), the central limit theoremsfor random sums have been studied for about 70 years. The central limittheorems for random sums of independent random variables play a very importantrole in various disciplines such as statistics, financial mathematics,insurance, etc. The purpose of this paper is to randomize some well-knownclassical and non-classical central limit theorems for sums of independent (notnecessarily identically distributed) random variables of a double sequence,with the conditions for determining their validity. The results obtained inthis paper are extensions and generalizations of known ones.

Classical and non-classical central limit theorems for random sums of independent random variables of a double sequence