An operatorial view on periodic correlated processes
by
Ilie Valusescu
Institute of Mathematics, Bucharest

Let E be a Hilbert space and H a right L(E)-module. If \Gamma:H × H --> L (E) is a correlation (see [1]), then {f_n} subset H is a stationary process if \Gamma[f_n, f_m]=\Gamma(m-n) is a function on m-n, and not by m and n separately.

In this paper a nonstationary process {f_n} subset H is considered, under the property that there exists a positive integer T such that

\Gamma[fn, fm]=\Gamma[fn+T, fm+T] = \Gamma(n, m) \leqno (1)

This correlation function depends on n and m separately and is a periodic one. A process {f_n} having the property (1) is called a periodic stationary process.

For each periodic stationary process there exists a unitary operator such that U f_n = f_n+T, so called the shift operator of {f_n}. Also for cross correlated periodic stationary processes there exists a common shift operator.

A prediction of the nonstationary process {f_n} can be made in a stationary way if for {f_n} a stationary process {X_n} subset H^T is attached in the following way

Xn = (fnT, fnT+1 , ... , f(n+1) T-1),

extending the correlation of L (E) on H to the correlation of L (E)^T to H^T, where T is the period of the process {f_n}.

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Date received: June 19, 2000