On some continuity and differentiability properties of paths of Gaussian processes

Stamatis Cambanis

Journal of Multivariate Analysis, 1973, vol. 3, issue 4, 420-434

Abstract: The following path properties of real separable Gaussian processes [xi] with parameter set an arbitrary interval are established. At every fixed point the paths of [xi] are continuous, or differentiable, with probability zero or one. If [xi] is measurable, then with probability one its paths have essentially the same points of continuity and differentiability. If [xi] is measurable and not mean square continuous or differentiable at every point, then with probability one its paths are almost nowhere continuous or differentiable, respectively. If [xi] harmonizable or if it is mean square continuous with stationary increments, then its paths are absolutely continuous with probability one if and only if [xi] is mean square differentiable; also mean square differentiability of [xi] implies path differentiability with probability one at every fixed point. If [xi] is mean square differentiable and stationary, then on every interval with probability one its paths are either differentiable everywhere or nondifferentiable on countable dense subsets. Also a class of harmonizable processes is determined for which of the following are true: (i) with probability one paths are either continuous or unbounded on every interval, and (ii) mean square differentiability implies that with probability one on every interval paths are either differentiable everywhere or nondifferentiable on countable dense subsets.

Keywords: Gaussian; processes; path; properties; stationary; processes; zero-one; laws (search for similar items in EconPapers)
Date: 1973
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Citations: View citations in EconPapers (2)

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