The book deals mainly with three problems involving Gaussian stationary
processes. The first problem consists of clarifying the conditions for
mutual absolute continuity (equivalence) of probability distributions of
a "random process segment" and of finding effective formulas for
densities of the equiva- lent distributions. Our second problem is to
describe the classes of spectral measures corresponding in some sense to
regular stationary processes (in par- ticular, satisfying the well-known
"strong mixing condition") as well as to describe the subclasses
associated with "mixing rate". The third problem involves estimation of
an unknown mean value of a random process, this random process being
stationary except for its mean, i. e., it is the problem of
"distinguishing a signal from stationary noise". Furthermore, we give
here auxiliary information (on distributions in Hilbert spaces,
properties of sam- ple functions, theorems on functions of a complex
variable, etc. ). Since 1958 many mathematicians have studied the
problem of equivalence of various infinite-dimensional Gaussian
distributions (detailed and sys- tematic presentation of the basic
results can be found, for instance, in [23]). In this book we have
considered Gaussian stationary processes and arrived, we believe, at
rather definite solutions. The second problem mentioned above is closely
related with problems involving ergodic theory of Gaussian dynamic
systems as well as prediction theory of stationary processes.
