Linearly-regular random process

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A stationary stochastic process (in the wide sense) , , for which the regularity condition

is satisfied, where is the mean square closed linear hull of the values , . (Here it is assumed that .) Regularity implies the impossibility of a (linear) prediction of the process in the very distant future; more precisely, if is the best linear prediction for with respect to the values , ,


A necessary and sufficient condition for regularity of a (one-dimensional) stationary process is the existence of a spectral density such that

The analytic conditions for regularity of multi-dimensional and infinite-dimensional stationary processes are more complicated. In the general case, when the spectral density is a positive operator function in Hilbert space, the regularity condition is equivalent to the fact that admits a factorization of the form

where , , is the boundary value of an operator function , , that is analytic in the lower half-plane , .

Every process that is stationary in the wide sense admits a decomposition into an orthogonal sum

where is a linearly-regular process and is a linearly-singular process, that is, a stochastic process that is stationary in the wide sense and for which


for all .


[1] Yu.A. Rozanov, "Stationary random processes" , Holden-Day (1967) (Translated from Russian)
[2] Yu.A. Rozanov, "Innovation processes" , Wiley (1977) (Translated from Russian)


One says more often purely non-deterministic process (in the wide sense) instead of linearly-regular process. The decomposition of a (second-order) process in a regular and a singular part (as in the main article) is known as the Wold decomposition.


[a1] I.A. Ibragimov, Yu.V. Linnik, "Independent and stationary sequences of random variables" , Wolters-Noordhoff (1971) (Translated from Russian)
[a2] J.L. Doob, "Stochastic processes" , Chapman & Hall (1953)
[a3] I.A. Ibragimov, Yu.A. Rozanov, "Gaussian random processes" , Springer (1978) (Translated from Russian)
How to Cite This Entry:
Linearly-regular random process. Yu.A. Rozanov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098