Vol. 20, No. 2, 1967

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Sub-stationary processes

Richard Mansfield Dudley

Vol. 20 (1967), No. 2, 207–215

This note supplements the longer paper [3]. It is proved in §2 that if T is a bounded Schwartz distribution on Rn, e.g. an L function, then its Fourier transform ϖT is of the form nf∕∂t1∂tn where f is integrable over any bounded set to any finite power. This follows from the main theorem of [3], but the proof here is much shorter.

Secondly, §3 shows that a p-sub-stationary random (Schwartz) distribution has sample distributions of bounded order. This generalizes a result of K. Ito for the stationary case.

Third, in §4 it is shown that p-sub-stationary stochastic processes define p-sub-stationary random distributions if p 1.

Mathematical Subject Classification
Primary: 60.50
Received: 10 December 1966
Published: 1 February 1967
Richard Mansfield Dudley