Multiplicity and representation theory of generalized random processes

Let (D) be the Schwartz space of infinitely differentiable scalar functions on the line, with compact supports, and ([Omega], [Sigma], P) be a fixed probability space. Let X : (D) --> L2([Omega], [Sigma], P) be a purely nondeterministic generalized random process (g.r.p.) in the sense of Itô with zero mean functional. A multiplicity representation theorem for X is obtained as a result of the Hellinger-Hahn theory. The representation can be assumed to be proper canonical. Thus each g.r.p. determines a unique cardinal number N <= [aleph]o, termed the multiplicity of the g.r.p. X. As a corollary, every stationary g.r.p. has multiplicity one. A class of harmonizable g.r.p.'s of multiplicity one can be constructed to include all the stationary g.r.p.'s. There exist g.r.p.'s of any prescribed multiplicity. The related linear least-squares prediction problem is obtained.