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.
[1]
D. Sarason.
Function theory on the unit circle
,
1978
.
[2]
K. Hoffman.
Banach Spaces of Analytic Functions
,
1962
.
[3]
N. Levinson,et al.
Weighted trigonometrical approximation onR1 with application to the Germ field of a stationary Gaussian noise
,
1964
.
[4]
一松 信,et al.
R.C. Gunning and H.Rossi: Analytic Functions of Several Complex Variables, Prentice-Hall, Englewood Cliffs, N.J., 1965, 317頁, 15×23cm, $12.50.
,
1965
.
[5]
D. Sarason,et al.
Past and Future
,
1967
.
[6]
W. Rudin,et al.
Extreme points and extremum problems in H1
,
1958
.