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Can a Gaussian process be almost surely discontinuous on an interval T but at the same time be continuous in Probability everywhere on T?

Alternative question: can a sequence of discontinuity points converge to a point where the Gaussian process is continuous?

Thank you very much for your input!

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1 Answer 1

Robert J. Adler's book An introduction to continuity, extrema, and related topics for general Gaussian processes has a lot of examples like this. In particular, on page 116 (exercise 6.1) he notes that if $(X_t)$ is Gaussian, and stationary with $\mathbb{E}|X_u-X_0|^2\sim 1/|\log(u)|$ the sample paths are discontinuous, though the process is continuous in probability.

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Thank you! My process is not stationary and I was hoping to find some kind of criterion to check things like that. –  Lena Sep 28 '12 at 16:30
    
@Lena Alder's book has loads of information on such processes, stationary and otherwise. –  Byron Schmuland Sep 28 '12 at 17:04

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