Characterizations of completely nondeterministic stochastic processes.

A discrete weakly stationary Gaussian stochastic process {x(t)}9 is completely nondeterministic if no non-trivial set from the σ-algebra generated by {x(t): t > 0} lies in the σ-algebra generated by {x(t): t ^ 0}. In [8] Levinson and McKean essentially showed that a necessary and sufficient condition for complete nondeterminism is that the spectrum of the process is given by \h\ where h is an outer function in the Hardy space, 77, of the unit circle in C with the property that h/h uniquely determines the outer function h up to an arbitrary constant. In this paper we consider several characterizations of complete nondeterminism in terms of the geometry of the unit ball of the Hardy space H and in terms of Hankel operators.