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Relative Weak Compactness of Sums of Random Variables

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Abstract

In this article sufficient conditions for relative weak compactness of sums centered by constants of pair-wise negatively associated randomvariables and for sums of squares of any random variables centered by their medians are given. These conditions become necessary and sufficient if random variables are independent. The conditions are inspired by classical conditions for weak convergence of sums of uniformly small random variables.

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References

  1. A. Araujo and E. Giné, The Central Limit Theoremfor Real and Banach Valued Random Variables (Wiley, New York, 1980).

    MATH  Google Scholar 

  2. M. Gerasimov, V. Kruglov, and A. Volodin, “On negatively associated random variables,” Lobachevskii J. Math. 33, 47–55 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  3. E. Giné, “Sums of independent random variables and sums of their squares,” Publ. Sec. Mat., No. 22, 127–132 (1980).

    Article  MathSciNet  Google Scholar 

  4. B. V. Gnedenko and A. N. Kolmogorov, Limits Distributions for Sums of Independent Random Variables, 2nd ed. (Addison-Wesley, Cambridge, MA, 1980).

    Google Scholar 

  5. C. R. Heathcote and J. W. Pitman, “An inequality for characteristic functions,” Bull. Austral. Math. Soc. 6 (1), 1–9 (1972).

    Article  MathSciNet  MATH  Google Scholar 

  6. V. M. Kruglov, “Relative weak compactness of sums of pair-wise independent random variables,” Statist. Probab. Lett. 122 (36), 36–41 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  7. M. Loéve, Probability Theory I, 4th ed. (Springer, New York, 1977).

    MATH  Google Scholar 

  8. G. Siegel, “Compactness of a sequence of sums of independent variables with values in a Hilbert space,” Lithuan. Math. J. 21, 331–341 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  9. A. V. Skorohod, Random Processes with Independent Increments (Kluwer Academic, Dordrecht, 1991).

    Book  Google Scholar 

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Author information

Authors and Affiliations

  1. Faculty of Physics and Mathematics, Moscow Region State University, ul. Radio 10a, Moscow, 105005, Russia

    N. N. Golikova

  2. Department of Statistics, Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow, 119992, Russia

    V. M. Kruglov

Authors
  1. N. N. Golikova
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  2. V. M. Kruglov
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Correspondence to N. N. Golikova.

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(Submitted by A. I. Volodin)

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Golikova, N.N., Kruglov, V.M. Relative Weak Compactness of Sums of Random Variables. Lobachevskii J Math 39, 331–339 (2018). https://doi.org/10.1134/S1995080218030137

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  • DOI: https://doi.org/10.1134/S1995080218030137

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