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.