Sufficient conditions for the continuity of stationary gaussian processes and applications to random series of functions
Annales de l'Institut Fourier, Volume 24 (1974) no. 2, p. 117-141

Let {X(t),t[0,1] n } be a stochastically continuous, separable, Gaussian process with E[X(t+h)-X(t)] 2 =σ 2 (|h|). A sufficient condition, in terms of the monotone rearrangement of σ, is obtained for X(t) 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 {X(t),t[0,1] n } un processus gaussien séparable et stochastiquement continu, satisfaisant à la condition E[X(t+h)-X(t)] 2 =σ 2 (|h|). On obtient une condition suffisante de continuité presque sûre de X(t), mise en termes de ré-arrangement monotone de σ. On fait l’application de ce résultat aux séries des fonctions aléatoires, en particulier, aux séries aléatoires de Fourier.

@article{AIF_1974__24_2_117_0,
     author = {Jain, Naresh C. and Marcus, Michael B.},
     title = {Sufficient conditions for the continuity of stationary gaussian processes and applications to random series of functions},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {24},
     number = {2},
     year = {1974},
     pages = {117-141},
     doi = {10.5802/aif.508},
     zbl = {0283.60041},
     mrnumber = {54 \#1356},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1974__24_2_117_0}
}
Jain, Naresh C.; Marcus, Michael B. Sufficient conditions for the continuity of stationary gaussian processes and applications to random series of functions. Annales de l'Institut Fourier, Volume 24 (1974) no. 2, pp. 117-141. doi : 10.5802/aif.508. http://www.numdam.org/item/AIF_1974__24_2_117_0/

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