Timeline for What structure is needed to define a Gaussian distribution on a given space?

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Mar 29, 2011 at 5:45	comment	added	Tom LaGatta		So the short answer is: it's not a restrictive assumption, in that there are LOTS of covariance functions for continuous Gaussian measures, though not every positive-definite function gives rise to one.
Mar 29, 2011 at 5:44	comment	added	Tom LaGatta		Well, a continuous Gaussian process on a compact parameter set $T$ (like the Wiener process on $[0,1]$) is exactly the realization of a Gaussian measure on the Banach space of continuous functions $C(T)$. Here, the covariance operator corresponds to the integral operator of the covariance function. Any positive-definite function serves as the covariance function to some measure, though there is a special class of them which corresponds to Gaussian processes, and a finer subclass which corresponds to continuous Gaussian processes.
Mar 28, 2011 at 23:33	comment	added	Simon Lyons		Thanks for your answer Tom. I don't have much intuition about Banach spaces and their duals. Is that a restrictive assumption? Can one always find such a measure $\mathbb{P}$?
Mar 28, 2011 at 15:32	history	answered	Tom LaGatta	CC BY-SA 2.5