Pseudo-regular and special functions and their application to problems of stochastic analysis

The aim of the project was a further development of the theory of pseudo-regular and special functions and their applications to limit theorems of renewal theory, study of asymptotic behaviour of solutions to stochastic and deterministic differential equations, investigation of linear and non-linear models in stochastic analysis, statistics of stochastic processes with long-term and short-term dependence, integral equations and transforms.
In the project:
the necessary and sufficient conditions of the convergence for series of autoregressive random variables and vectors as well as the necessary and sufficient conditions under which the strong law of large numbers holds true are obtained (the results are published in the form of monograph);
exact values of subgaussian norms for binary distributions are found; this allowed us to obtain large deviations inequalities for sums of binary random variables in an ultimate unimprovable form;
new important generalizations of hypergeometric and Legendre functions are introduced; this allowed us to examine new kinds of integral transforms and to apply them for solving new differential and integral equations of mathematical physics (the results are included into a monograph);
new significant results in the theory of pseudo-regular functions are obtained and their applications to the study of the asymptotic behaviour of various stochastic processes are discussed (the applications range from generalized renewal processes to solutions of stochastic differential equations) (the results are included into a monograph);
new limit properties of multidimensional integrals with cyclic kernels are derived; this allows us to obtain the conditions of the asymptotic normality for correlogram estimates of impulse response functions for unstable Volterra systems with intrinsic noises;
the conditions of the convergence for generalized Spitzer series with pseudo-regular functions are obtained;
conditions for the strong law of large numbers for stochastic processes with quasi-additive moment functions are found.