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The modern theory of subgaussian random variables and processes was created by independent efforts of several research schools from France, USA and Ukraine. Professor Yu.Kozachenko was a founder and leading figure of this research direction of the Ukrainian probability school. An outline of Professor Yu.Kozachenko's contribution to the theory of sub-Gaussian random variables and processes is presented. The class of $\varphi$-subgaussian random variables is introduced and its key property is discussed. Then it is demonstrated how these results can be used in stochastic approximation and modeling. In particular, applications to approximation of trajectories of $\varphi$-subgaussian random processes with given accuracy and reliability are discussed. Two important clases of algorithms from the signal processing theory, the Shannon sampling method and wavelet decompositions, are used as examples. Some personal memories of the author about Yu. Kozachenko are included at the end of the paper.

Sharp upper bounds for interpolation remainders of the multidimensional Paley-Wiener class functions by finite regular Whittaker-Kotel’nikov-Shannon sampling sum are obtained. The extremal functions are given for which the derived bounds are attained. Truncation error analysis and convergence rate is provided in weak Cramer class random fields. The historical background, the development, and extensive reference list are given concerning truncation error upper bounds for deterministic and random signal functions. Finally, new research directions are posed and discussed.

Time shifted aliasing error upper bound extremals for the sampling reconstruction procedure are fully characterized. Sharp upper bounds are found on the aliasing error of truncated cardinal series and the corresponding extremals are described for entire functions from certain specific L p , p > 1 , classes. Analogous results are obtained in multidimensional regular sampling. Truncation error analysis is provided in all cases considered. Moreover, sharpness of bounding inequalities, convergence rates and various sufficient conditions are discussed.

Characterization of all extremals for uniform upper bound on aliasing error is given in regular Whittaker-Kotel’nikov-Shannon sampling procedure. Sharp uniform upper bounds on aliasing error of truncated sampling cardinal series and corresponding extremals are obtained. Truncation error analysis and convergence rate are given.

New sharp upper bound on the magnitude of √x J ν(x) is derived. The comparison with bounds appearing in earlier works is made. Bounds on |J ν(x)−√(2/π x)cos (x−πν/2−π/4)| are obtained and discussed. An application to conditionally convergent Weber–Schafheitlin type integrals is shown.

Homogeneous in the wide sense, covariant random fields on commutative local compact groups with values in finite-dimensional complex Hilbert spaces are considered. The general formula for the correlation operator of such a field is proved, as well as the spectral representation of the field itself in the form of a series of stochastic integrals with respect to orthogonal random measures.