Far Eastern Mathematical Journal

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Two sided bounds of rate convergence in limit theorem for minimum of random vectors


G. Sh. Tsitsiashvili

2005, issue 1-2, Ń. 82–87


Abstract
In this article upper and low bounds of a rate convergence for minimums of independent and identically distributed random (i.i.d.r.) vectors are constructed. These bounds have common power and different logarithmical multiplyers. An interest to this problem is called by following causes. At first I. Siganov obtained upper bounds for minimums of i.i.d.r. variables, which may be considered as a foundation for two sided bounds. At second last years P. Rocchi constructed new models of a life-time for biological objects, which are based on stochastic entropy methods and give distributions analogous to considered ones. At third in mathematical statistics and reliability theory there are so called Marshall-Olkin distributions, which may be interpreted as limit distributions for minimums of i.i.d.r. vectors. This interpretation widens a class of Marshall-Olkin distributions.

Keywords:
limit distributions for minimums of random vectors, upper and low bounds of rate convergence

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References

[1] I. S. Siganov, “Several remarks on applications of one approach to studies of characterization problems of Polya theorem type”, Proceedings of the 6-th International Seminar, Lecture Notes in Mathematics, 1983, 227–237.
[2] P. Rocchi, “Boltzman-like Entropy in Reliability Theory”, Entropy, 4 (2002), 142–150.
[3] P. Rocchi, G. Sh. Tsitsiashvili, “About the Reversibility and Irreversibility of Stochastic Systems”, Proceedings of International Conference on Foundations of Probability and Physics-3, Vaxjo University, Sweden, 2004 (to appear).
[4] E. J. Gumbel, “Bivariate exponential distributions”, J. Amer. Statist. Assoc., 55:292 (1960), 698–707.
[5] A. W. Marshall, I. Olkin, “A multivariate exponential distribution”, J. Amer. Statist. Assoc., 62:317 (1967), 30–44.
[6] J. E. Freund, “A bivariate extension of the exponential distribution”, J. Amer. Statist. Assoc., 56:296 (1961), 971–977.

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