Valeriĭ V. Buldygin

Properties of a Subclass of Avakumović Functions and Their Generalized Inverses

We study properties of a subclass of ORV functions introduced by Avakumović and provide their applications for the strong law of large numbers for renewal processes.

The sub-Gaussian norm of a binary random variable

The exact values of the sub-Gaussian norms of Bernoulli random variables and binary random variables are found. Exponential bounds for the distributions of sums of centered binary random variables are studied for both cases of independent and dependent random variables. These bounds improve some known results.

On some properties of asymptotic quasi-inverse functions

A characterization of normalizing functions connected with the limiting behavior of ratios of asymptotic quasi-inverse functions is discussed. For nondecreasing functions, conditions are obtained that are necessary and sufficient for their asymptotic quasi-inverse functions to belong to the class of (so-called) O-regularly varying functions or to some of its subclasses.

PRV property of functions and the asymptotic behaviour of solutions of stochastic differential equations

In this paper, we investigate the a.s. asymptotic behaviour of the solution of the stochastic differential equation dX(t) = g(X(t)) dt+σ(X(t)) dW (t), where g(·) and σ(·) are positive continuous functions and W (·) is a standard Wiener process. By an application of the theory of PRV and PMPV functions, we find conditions on g(·) and σ(·), under which X(·) may be approximated a.s. on {X(t) → ∞} by the solution of the deterministic differential equation dμ(t) = g(μ(t)) dt. Moreover, we study the asymptotic stability with respect to initial conditions of solutions of the above SDE as well as the asymptotic behaviour of generalized renewal processes connected with this SDE.

Inequalities for the distributions of functionals of sub-Gaussian vectors

Exponential inequalities for moment generating functions and for distributions of sub-Gaussian random vectors are studied in the paper.

A generalization of Karamata’s theorem on the asymptotic behavior of integrals

A generalization of Karamata’s theorem on the asymptotic behavior of integrals of regularly varying functions with oscillating components is obtained in the paper.

Generalized Renewal Processes

This chapter is organized as follows. In Sect. 8.2, generalized renewal processes constructed from random sequences are studied and a number of results are proved on the relationship between the SLLN for the underlying random sequence and the SLLN for the corresponding renewal process. These results explain the nature of the SQI-, PRV-, and POV-properties of the normalizing sequences under rather simple conditions, for which these properties are satisfied. Generalized renewal processes constructed from stochastic processes are studied in Sect. 8.4. The results of this section also explain the role of the SQI-, PRV-, and POV-properties of the normalizing sequences in the SLLN for renewal processes. In addition, we make clear, why the requirement of continuity of the paths of the underlying process is important. The results of Sects. 8.2 and 8.4 are used in Sects. 8.3 and 8.5, respectively. These sections contain a discussion of several particular questions.

Asymptotic Behavior of Solutions of Stochastic Differential Equations

This chapter aims at finding nonrandom approximations (a precise definition is given below) of solutions of a general class of stochastic differential equations. We follow the setting by Gihman and Skorohod [149], however the results below are more general.

Nondegenerate Groups of Regular Points

The defining property of an ORV-function f is that \(f\in \mathbb {{F}}_{+}\) is measurable and the upper limit function exists and is positive and finite (see Definition 3.7). The main aim of this chapter is to study a subclass of functions in ORV with “nondegenerate group of regular points”, that is, those ORV-functions for which a limit function exists (see Definition 3.2) and is positive and finite belonging to a certain multiplicative subgroup in R+.

Equivalence of Limit Theorems for Sums of Random Variables and Renewal Processes

Many limit results are known for cumulative sums of independent identically distributed random variables and the corresponding renewal counting processes to hold under the same conditions. Despite the coincidence of conditions for both cases, the theories have been developed independently of each other and the methods are different. A general question is whether or not the whole theories of limit theorems for sums and renewal processes are equivalent. In this chapter, we consider the problem of the equivalence of certain asymptotic results, like the strong law of large numbers or the law of the iterated logarithm, for a sequence of sums of independent, identically distributed random variables and its corresponding renewal process.