Theorems of large deviations for the sums of a random number of independent random variables

In probability theory, the topic of large deviations, i. e., approximation problems of the probabilities of rare events, have a significant place. To understand why rare events are important at all one only has to think of the events in an insurance mathematics, nuclear physics and etc., to be convinced that those events can have an enormous impact.

This thesis is concerned with a normal approximation to a distribution of the sum ZN = PN j=1 ajXj ; Z0 = 0, 0 < aj < 1, of a random number of summands N of independent identically distributed weighted random variables fX;Xj ; j = 1; 2; :::g that takes into consideration large deviations in both the Cramér zone (the characteristic functions of the summands of ZN are analytic in a vicinity of zero) and the power Linnik zone (the growth of the moments of the summands does not ensure the analyticity of the characteristic functions). Here a non-negative integer-valued random variable N is independent of fX;Xj ; j = 1; 2; :::g. In addition, the asymptotic expansion that take into consideration large deviations in the Cramér zone for the density function of the standardized compound Poisson process is obtained. To solve the problems, the classical method of characteristic functions, cumulant and combinatorial methods are used.

Although, in probability theory the asymptotic behavior of tail probabilities for the sums of a random number of summands of random variables is a quite new problem, it was initiated in the XXth century, but there is a very extensive literature on mentioned problem. However, as it is known for the author of the dissertation, there are a few scientific works on theorems of large deviations for the sums of a random number of summands of independent random variables in case where the cumulant method is used.

The thesis consists of an introduction, three chapters, general conclusions, references, and a list of the author’s publications. The introduction reveals the importance of the scientific problem, describes the tasks of the thesis, research methodology, scientific novelty, the practical significance of results. In the first chapter an overview of the problems is presented. The second chapter is devoted for obtaining an upper bound for the cumulants, theorems of large deviations and exponential inequalities for the standardized version of the sum ZN. The instances of large deviations (the law of N is known; aj 1; discount version of large deviations) are also analyzed in this chapter. In the third chapter, the asymptotic expansion of large deviations in the Cramér zone for the density function of the standardized compound Poisson process is considered.

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DOI: https://doi.org/10.20334/2202-M