Sufficient conditions for the continuity of stationary gaussian processes and applications to random series of functions

Jain, Naresh C.; Marcus, Michael B.

Annales de l'Institut Fourier, Tome 24 (1974) no. 2, pp. 117-141.

Let $\left\{X\right(t),t\in [0,1{]}^n\}$ be a stochastically continuous, separable, Gaussian process with $E\left[X\right(t+h)-X(t){]}^2={\sigma }^2(\left|h\right|)$. A sufficient condition, in terms of the monotone rearrangement of $\sigma$, is obtained for $X\left(t\right)$ to have continuous sample paths almost surely. This result is applied to a wide class of random series of functions, in particular, to random Fourier series.

Soit $\left\{X\right(t),t\in [0,1{]}^n\}$ un processus gaussien séparable et stochastiquement continu, satisfaisant à la condition $E\left[X\right(t+h)-X(t){]}^2={\sigma }^2(\left|h\right|)$. On obtient une condition suffisante de continuité presque sûre de $X\left(t\right)$, mise en termes de ré-arrangement monotone de $\sigma$. On fait l’application de ce résultat aux séries des fonctions aléatoires, en particulier, aux séries aléatoires de Fourier.

[1] R. P. Boas Jr. and M. B. Marcus, Inequalities involving a function and its inverse, SIAM J. Math. Anal., 4 (1973). | MR | Zbl

[2] Dudley R. M., The sizes of compact subsets of Hilbert space and continuity of Gaussian processes, J. Functional Analysis, 1 (1967), 290-330. | MR | Zbl

[3] R. M. Dudley, Sample functions of the Gaussian process, Ann. of Probability, 1 (1973), 66-103. | MR | Zbl

[4] X. Fernique, Continuité des processus Gaussiens, C.R. Acad. Sci. Paris, 258 (1964), 6058-6060. | MR | Zbl

[5] G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, Cambridge Univ. Press, (1934), Cambridge, England. | JFM | Zbl

[6] N. C. Jain, Conditions for the continuity of sample paths of a Gaussian process, unpublished manuscript, (1972).

[7] N. C. Jain and G. Kallianpur, A note on the uniform convergence of stochastic processes, 41 (1970), 1360-1362. | MR | Zbl

[8] J. P. Kahane, Some random series of functions, (1968), D. C. Heath, Lexington, Mass. | MR | Zbl

[9] E. Lukacs, Characteristic functions, Second Edition, (1970), Hafner, New York. | MR | Zbl

[10] M. B. Marcus, A comparaison of continuity conditions for Gaussian processes., Ann. of Probability, 1 (1973), 123-130. | MR | Zbl

[11] M. B. Marcus, Continuity of Gaussian processes and random Fourier series, Ann. of Probability, 1 (1973), 968-981. | MR | Zbl

[12] M. B. Marcus and L. A. Shepp, Continuity of Gaussian processes., Trans. Amer. Math. Soc., 151 (1970), 377-392. | MR | Zbl

[13] J. L. Doob, Stochastic processes, (1953), John Wiley and Sons, New York. | MR | Zbl