Sub-stationary processes

Richard Mansfield Dudley

Vol. 20 (1967), No. 2, 207–215

DOI: 10.2140/pjm.1967.20.207

Abstract

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⋯∂t_n 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

Milestones

Received: 10 December 1966

Published: 1 February 1967