From subgaussianity to stochastic approximation and modelling

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 '-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 '-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.