Linearly-regular random process

A stationary stochastic process (in the wide sense) $ \xi ( t) $, $ - \infty < t < \infty $, for which the regularity condition

$$ \cap _ { t } H _ \xi ( - \infty , t ) = 0 $$

is satisfied, where $ H _ \xi ( - \infty , t ) $ is the mean square closed linear hull of the values $ \xi ( s) $, $ s \leq t $. (Here it is assumed that $ {\mathsf E} \xi ( t) = 0 $.) Regularity implies the impossibility of a (linear) prediction of the process $ \xi ( t) $ in the very distant future; more precisely, if $ \widehat \xi ( t + u ) $ is the best linear prediction for $ \xi ( t + u ) $ with respect to the values $ \xi ( s) $, $ s \leq t $,

$$ {\mathsf E} | \xi ( t + u ) - \widehat \xi ( t + u ) | ^ {2} = \ \min _ {\eta \in H _ \xi ( - \infty , t ) } \ {\mathsf E} | \xi ( t + u ) - \eta | ^ {2} , $$

then

$$ \lim\limits _ {u \rightarrow \infty } \widehat \xi ( t + u ) = 0 . $$

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

$$ \int\limits _ {- \infty } ^ \infty \frac{ \mathop{\rm ln} f ( \lambda ) }{1 + \lambda ^ {2} } d \lambda > - \infty . $$

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

$$ f ( \lambda ) = \phi ^ {*} ( \lambda ) \phi ( \lambda ) , $$

where $ \phi ( \lambda ) $, $ - \infty < \lambda < \infty $, is the boundary value of an operator function $ \phi ( \lambda + i \mu ) $, $ \mu \rightarrow 0 $, that is analytic in the lower half-plane $ z = \lambda + i \mu $, $ \mu < 0 $.

Every process $ \zeta ( t) $ that is stationary in the wide sense admits a decomposition into an orthogonal sum

$$ \zeta ( t) = \xi ( t) + \eta ( t) , $$

$$ {\mathsf E} \xi ( t) \overline{ {\eta ( t) }}\; = 0 , $$

where $ \xi ( t) $ is a linearly-regular process and $ \eta ( t) $ is a linearly-singular process, that is, a stochastic process that is stationary in the wide sense and for which

$$ \cap _ { t } H _ \eta ( - \infty , t ) = H _ \eta ( - \infty , \infty ) ; $$

also,

$$ H _ \xi ( - \infty , t ) \subset H _ \zeta ( - \infty , t ) \ \ \textrm{ and } \ H _ \eta ( - \infty , t ) \subset H _ \zeta ( - \infty , t) $$

for all $ t $.

References

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

Comments

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.

References

[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)