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June 23, 2021, at the Dissertation Council D 003.015.01 at the Sobolev Institute of Mathematics SB RAS Evgeny I. Shefer defended his candidate's thesis "Asymptotic analysis of the distribution of the sojourn time of a random walk in the region of moderately large deviations". The dissertation was carried out under the scientific supervision of the doctor of physics and mathematics, professor Igor S. Borisov, on specialty 01.01.05 鈥� Probability theory and mathematical statistics.

The main results of the thesis are devoted to the asymptotic analysis of the distribution of the random variable 蟿n (xg) 鈥� the sojourn time of the trajectory of the classical random walk S1, ..., Sn above the level xg () for a sufficiently wide class of functions g () defining the configuration of the boundary level, provided that x = x (n) as n 鈫� 鈭� increases indefinitely at a rate corresponding to the zone of moderately large deviations. Under the Cramer condition on the distribution of the jump of a random walk, exact asymptotic relations are obtained as n 鈫� 鈭� for the expectation E蟿n (xg), and in the case g (路) 鈮� 1 and for the tail of the distribution P {蟿n (x) 鈮� y} for any fixed y.

In addition, a number of estimates are obtained for the rate of convergence in the arcsine law for the classical random walk. Moreover, for the simplest symmetric random walk, these estimates are unimprovable, which made it possible to obtain an asymptotic representation for the tail of the distribution of the sojourn time of the trajectory of the indicated random walk above the receding rectilinear level x with a velocity corresponding to the entire range of moderately large deviations. For a more general random walk, the corresponding asymptotic representation was proved for a slightly narrower zone of deviations. The question of extending the latter result to the entire zone of moderately large deviations is still open.

As a promising study, we can single out the search for the asymptotic behavior of the tail of the distribution P {蟿n (xg) 鈮� y} in the case of a smooth curvilinear boundary, which is determined by the function g(路).