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Title:  Theorems of Large Deviations for the Sums of a Random Number of Independent Random Variables 
Other Titles:  Atsitiktinio skaičiaus nepriklausomų dėmenų sumos didžiųjų nuokrypių teoremos 
Authors:  KASPARAVIČIŪTĖ, Aurelija 
Issue Date:  2013 
Publisher:  VGTU leidykla „Technika" 
Citation:  Kasparavičiūtė, A. 2013. Theorems of Large Deviations for the Sums of a Random Number of Independent Random Variables: doctoral dissertation. Vilnius: Technika, 128 p. 
Description:  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 nonnegative integervalued 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. 
URI:  http://dspace1.vgtu.lt/handle/1/1670 
ISBN:  9786094575976 
Appears in Collections:  Fizinių mokslų daktaro disertacijos ir jų santraukos

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