View source for Linearly-regular random process

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A [[Stationary stochastic process|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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.A. Rozanov,   "Stationary random processes" , Holden-Day  (1967)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  Yu.A. Rozanov,   "Innovation processes" , Wiley  (1977)  (Translated from Russian)</TD></TR></table>

====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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.A. Ibragimov,   Yu.V. Linnik,   "Independent and stationary sequences of random variables" , Wolters-Noordhoff  (1971)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.L. Doob,   "Stochastic processes" , Chapman &amp; Hall  (1953)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  I.A. Ibragimov,   Yu.A. Rozanov,   "Gaussian random processes" , Springer  (1978)  (Translated from Russian)</TD></TR></table>