Geodesic
Criteria of relative and stochastic compactness for distributions of sums of independent random variables
Teoriâ veroâtnostej i ee primeneniâ, Tome 63 (2018) no. 1, pp. 70-88 Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: sums of independent random variables, scheme of series, relative compactness, stochastic compactness, characteristic functions. 
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A. A. Khartov. Criteria of relative and stochastic compactness for distributions of sums of independent random variables. Teoriâ veroâtnostej i ee primeneniâ, Tome 63 (2018) no. 1, pp. 70-88. http://geodesic.mathdoc.fr/item/TVP_2018_63_1_a3/

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