Best approximation for classes of random processes

Alexander Buslaev and Oleg Seleznjev

Department of Mathematical Statistics,
Lund Institute of Technology,
Lund University,

ISSN 0281-1944

We consider the best approximation order for classes of random processes. The close relationship between the smoothness properties of a function and the best rate of its linear approximation is one of the basic ideas of conventional (deterministic) approximation theory. We investigate similar properties for random functions. Hölder's class of random processes is studied in more detail. Several types of extremal characteristics of an approximation accuracy are proposed. For Hölder's class of random processes, the exact order of these characteristics are found in different metrics. These results can be used, for example, for a justification of a specific approximation method selection in approximation problems.
Key words:
approximation, random process, Gaussian process, Kahrunen-Loève expansion, n-width