Regularity of Gaussian processes. (English) Zbl 0712.60044

Talagrand, Michel

Acta Math. 159, No. 1-2, 99-149 (1987).

The author obtains necessary and sufficient conditions for the continuity or boundedness of a Gaussian process. A (centered) Gaussian process is a family \((X_ t)_{t\in T}\) of real-valued random variables, indexed by some index set T, such that every finite linear combination \(\sum a_ tX_ t\) is a real-valued Gaussian random variable. The covariance function \(\Gamma (u,v)=E(X_ uX_ v)\) on \(T\times T\) determines E(\(\sum a_ tX_ t)^ 2\) and hence determines the law of the process \((X_ t)_{t\in T}\). The problem of obtaining necessary and sufficient conditions for the continuity and/or boundedness of a Gaussian process in terms of its covariance is attributed to Kolmogorov. In the case of stationary Gaussian processes, i.e. when \(\Gamma (u,v)=\Gamma (u-v,0)\), this problem was solved in the beautiful theorem of R. M. Dudley [J. Funct. Anal. 1, 290-330 (1967; Zbl 0188.205)] and X. Fernique [Lect. Notes Math. 480, 1-96 (1975; Zbl 0331.60025)].
Suppose that the index set T is a compact pseudometric space with respect to the pseudodistance \(d=(E(X_ u-X_ v)^ 2)^{1/2}\) and denote by \(N_{\epsilon}\) the smallest number of closed d-balls of radius \(\epsilon\) that cover T. Then the stationary Gaussian process \((X_ t)_{t\in T}\) has a version with continuous sample paths almost surely if and only if \(\int^{\infty}_{0}(\log N_{\epsilon})^{1/2}d\epsilon <\infty\). However, it has been known for a long time that this condition is not necessary if the Gaussian process is not stationary.
X. Fernique [C. R. Acad. Sci., Paris, Sér. A 278, 363-365 (1974; Zbl 0274.60029)] showed that if there exists a probability measure m on (T,d) such that \[ (1)\quad \sup_{t\in T}\int^{\infty}_{0}(\log 1/m(B(t,\epsilon)))^{1/2}d\epsilon <\infty, \] where B(t,\(\epsilon\)) denotes the d-ball of radius \(\epsilon\) centered at t, then the Gaussian process determined by d has bounded sample paths almost surely. This result was implicit in earlier work by Preston based on an important result of Garcia, Rodemich and Rumsey. (This paper contains a good historical discussion and references to these and other important papers.) It is remarkable that the author is able to show that (1) is necessary. To be more explicit, he shows that for each bounded Gaussian process \((X_ t)_{t\in T}\) there exists a probability measure m on (T,d) such that \[ \sup_{t\in T}\int^{\infty}_{0}(\log 1/m(B(t,\epsilon)))^{1/2}d\epsilon \leq K E\sup_{t\in T} X_ t, \] for some universal constant K. Closely related results give necessary and sufficient conditions for continuity.
There are several important unexpected consequences of the methods used by the author and many interesting applications. For example here is his Theorem 15: Let \((X_ t)_{t\in T}\) be a Gaussian process and \((Y_ t)\) be any other centered process indexed by the same set. Assume that for each \(\theta\) in \({\mathbb{R}}\), we have \[ E \exp \theta (Y_ u-Y_ v)\leq E \exp \theta (X_ u-X_ v)=\exp (\theta^ 2d^ 2(u,v)/2). \] Then we have E \(\sup_{t\in T}Y_ t\leq K E \sup_{t\in T}X_ t\), where K is a universal constant. This is a deep and important paper. An understanding of it is essential for any further serious research in Gaussian processes.

MSC:

60G17	Sample path properties
60G15	Gaussian processes
60G12	General second-order stochastic processes

Keywords:

continuity or boundedness of a Gaussian process; stationary Gaussian processes; historical discussion

Citations:

Zbl 0188.205; Zbl 0331.60025; Zbl 0274.60029

References:

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