Vol. 107, No. 2, 1983

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Characterizations of completely nondeterministic stochastic processes

Peter Bloomfield, Nicolas P. Jewell and Eric Hayashi

Vol. 107 (1983), No. 2, 307–317

A discrete weakly stationary Gaussian stochastic process {x(t)}, 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|2 where h is an outer function in the Hardy space, H2, 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 H1 and in terms of Hankel operators.

Mathematical Subject Classification 2000
Primary: 60G25
Secondary: 60G10
Received: 30 October 1981
Published: 1 August 1983
Peter Bloomfield
Nicolas P. Jewell
Eric Hayashi