TY - JOUR

T1 - Criteria of relative and stochastic compactness for distributions of sums of independent random variables

AU - Khartov, A. A.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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.

AB - 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.

KW - Characteristic functions

KW - Relative compactness

KW - Scheme of series

KW - Stochastic compactness

KW - Sums of independent random variables

UR - http://www.scopus.com/inward/record.url?scp=85064689639&partnerID=8YFLogxK

U2 - 10.1137/S0040585X97T988915

DO - 10.1137/S0040585X97T988915

M3 - Article

AN - SCOPUS:85064689639

VL - 63

SP - 57

EP - 71

JO - Theory of Probability and its Applications

JF - Theory of Probability and its Applications

SN - 0040-585X

IS - 1

ER -