Some properties and examples of random processes that are almost wide sense stationary

A wide sense stationary (WSS) process
X(t), t \in I
, has shift operators
T_h: X(t) \rightarrow X(t + h), h \in I
, which are unitary operators in the Hilbert space
H(X)
generated in the usual way by
X(t)
. We study the class of uniformly hounded linearly stationary (UBLS) processes; This is the class of processes having shift operators
T_h
that are linear and bounded with
\parallel T_h \parallel ^ 2 \leq M
, for some constant
M
. Examples are given of UBLS processes resulting from linear transformations on non-stationary white noise. The notion of an UBLS almost white noise process is defined, and some special cases are studied. Also, possible applications to time series modeling are indicated. The canonical structure of a finite-dimensional deterministic UBLS process is obtained. Theorems for superposition and multiplication of UBLS processes are presented. Finally, continuous-time white noise is given a rigorous treatment in terms of generalized processes, and conditions for UBLS are given.