Criteria of relative and stochastic compactness for distributions of sums of independent random variables

We consider sequences of distributions of centered sums of independent random variables within the scheme of series without imposing the classical conditions of uniform asymptotic negligibility and uniform asymptotic constancy. A criterion of relative compactness for such sequences of distributions was obtained by Siegel [Lith. Math. J., 21 (1981), pp. 331–341]. In the present paper this criterion is formulated in a more complete form, and a new proof is proposed based on characteristic functions. We also obtain a criterion of stochastic compactness, which is a stronger property than the one introduced by Feller [Proc. 5th Berkeley Symposium on Mathematical Statistics and Probability, Berkeley, CA, 1965/66, Vol. 2: Contributions to Probability Theory, Part 1, 1967, pp. 373–388]. Moreover, several new criteria of relative and stochastic compactness for such sequences of distributions are proposed in terms of characteristic functions of summable random variables.