The article explores in detail higher education studies that appear as one of the essential university processes. University studies are not a new phenomenon; however, they are overwhelmed by the volume of information surrounded by a wide range of diverse stakeholders. Therefore, the university inevitably needs changes in the adequate fulfillment of its mission thus meeting and harmonizing the expectations of different stakeholders of modern society and the state. Therefore, the concept and role of the study process itself in society are changing. The studies considered to be timely and qualitative are becoming a more and more relevant question to universities. Based on the previous scientific analysis of the study process at the university level and a concept of Quality Assurance for university studies formulated by the Bologna Process, the article examines the relationships and importance of the components (criteria) composing the study process at the university level. The article is aimed at revealing the diversity of the study process and at evaluating the importance and significance of the criteria composing it. To achieve this goal, the multi-criteria decision-making method the Analytical Hierarchy Process were invoked. The representatives of two major Lithuanian universities participated in the carried out research the results of which demonstrated that the criteria of the study process were fundamentally different, and some of those were difficult to measure applying quantitative parameters. Despite this circumstance, giving more attention to a combination of criteria for a particular process of university studies creates conditions for purposeful modeling the study process and the pursuit of high-quality university studies.

In this paper, we consider a random variable Z t = Nt i=1 a i X i , where X, X 1 , X 2 , . . . are independent identically distributed random variables with mean EX = μ and variance DX = σ 2 > 0. It is assumed that Z 0 = 0, 0 ≤ a i < ∞, and N t , t ≥ 0 is a non-negative integervalued random variable independent of X i , i = 1, 2, . . . . The paper is devoted to the analysis of accuracy of the standard normal approximation to the sumZ t = (DZ t ) −1/2 (Z t − EZ t ), large deviation theorems in the Cramer and power Linnik zones, and exponential inequalities for P(Z t ≥ x).

In this paper, we present the rate of convergence of normal approximation and the theorem on large deviations for a compound process Zt = \sumNt i=1 t aiXi, where Z0 = 0 and ai > 0, of weighted independent identically distributed random variables Xi, i = 1, 2, . . . with  mean EXi = µ and variance DXi = σ2 > 0. It is assumed that Nt is a non-negative integervalued random variable, which depends on t > 0 and is independent of Xi, i = 1, 2, . . . .

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 Z N = N j=1 a j X j , Z 0 = 0, 0 < a j < ∞, of a random number of summands N of independent identically distributed weighted random variables {X, X j , j = 1, 2, ...} that takes into consideration large deviations in both the Cramér zone (the characteristic functions of the summands of Z N 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 {X, X j , j = 1, 2, ...}. 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 Z N. The instances of large deviations (the law of N is known; a j ≡ 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. i. i. d.-independent identically distributed; r. n. s.-random number of summands; a. s.-almost sure.

In the present paper we consider weighted random sums ZN = ∑j=1NajXj, where 0 ≤ aj < ∞, N denotes a non-negative integer-valued random variable, and {X, Xj , j = 1, 2,...} is a family of independent identically distributed random variables with mean EX = µ and variance DX = σ2 > 0. Throughout this paper N is independent of {X, Xj , j = 1, 2,...} and, for definiteness, it is assumed Z0 = 0. The main idea of the paper is to present results on theorems of large deviations both in the Cramér and power Linnik zones for a sum ~ZN = (ZN − EZN )(DZN )−1/2 , exponential inequalities for a tail probability P(~ZN > x) in two cases: µ = 0 and µ ≠ 0 pointing out the difference between them. Only normal approximation is considered. It should be noted that large deviations when µ ≠ 0 have been already considered in our papers [1,2].