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WROCŁAW UNIVERSITY
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Contents of PMS, Vol. 5, Fasc. 1,
pages 45 - 58
 

ON RECURRENT DIFFERENTIAL REPRESENTATIONS FOR STATIONARY STOCHASTIC PROCESSES

Lesław Bielak

Abstract: In this paper differential representations for stationary stochastic processes with quotients of analytic functions of minimal type as spectral characteristics are given. Such a process is a limit (in the mean square sense) of stationary stochastic processes y (t) n (n = 1,2,...) which are solutions of an infinite-dimensional system of stochastic differential equations. There are some recurrent connections between y (t) n and for that reason we call the differential representations considered in this paper recurrent. The representations are applied to find a necessary and sufficient condition for absolute Continuity of measures generated by Gaussian stationary processes with spectral characteristics mentioned above. This condition takes the form

gy(c) cli-->m oo gx(c) = 1.
Thereby the Feldman theorem is generalized.

2000 AMS Mathematics Subject Classification: Primary: -; Secondary: -;

Key words and phrases: -

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