Vol. 97, No. 1, 1981

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Sample functions of Pólya processes

Takayuki Kawada

Vol. 97 (1981), No. 1, 125–135
Abstract

For a nonnegative measurable function f satisfying

∫
∞
−∞f (x) dx = 1,

define

      ∫ ∞
r(t) =     min{f(x),f(x + t)} dx.
−∞

Berman proved, extending so-called “Polya characteristic function”, that the r is the characteristic function of an absolutely continuous distribution. The positive-definiteness of the r corresponds to a stationary Gaussian process, which is called Polya-Covariance process or simply Polya process.

In this paper, some analytic properties of its sample functions are studied: (1) continuity, (2) differentiability, (3) quadratic variation, and (4) upper and lower class.

Mathematical Subject Classification 2000
Primary: 60G15
Secondary: 60G17
Milestones
Received: 29 July 1980
Published: 1 November 1981
Authors
Takayuki Kawada