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

• Volume: 24, Issue: 2, page 117-141
• ISSN: 0373-0956

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## Abstract

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Let $\left\{X\left(t\right),\phantom{\rule{0.277778em}{0ex}}t\in \left[0,1{\right]}^{n}\right\}$ be a stochastically continuous, separable, Gaussian process with $E\left[X\left(t+h\right)-X\left(t\right){\right]}^{2}={\sigma }^{2}\left(|h|\right)$. 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.

## How to cite

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Jain, Naresh C., and Marcus, Michael B.. "Sufficient conditions for the continuity of stationary gaussian processes and applications to random series of functions." Annales de l'institut Fourier 24.2 (1974): 117-141. <http://eudml.org/doc/74167>.

@article{Jain1974,
abstract = {Let $\lbrace X(t),\; t\in [0,1]^n\rbrace$ be a stochastically continuous, separable, Gaussian process with $E[X(t+h)-X(t)]^2=\sigma ^2(\vert h\vert )$. A sufficient condition, in terms of the monotone rearrangement of $\sigma$, 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.},
author = {Jain, Naresh C., Marcus, Michael B.},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {2},
pages = {117-141},
publisher = {Association des Annales de l'Institut Fourier},
title = {Sufficient conditions for the continuity of stationary gaussian processes and applications to random series of functions},
url = {http://eudml.org/doc/74167},
volume = {24},
year = {1974},
}

TY - JOUR
AU - Jain, Naresh C.
AU - Marcus, Michael B.
TI - Sufficient conditions for the continuity of stationary gaussian processes and applications to random series of functions
JO - Annales de l'institut Fourier
PY - 1974
PB - Association des Annales de l'Institut Fourier
VL - 24
IS - 2
SP - 117
EP - 141
AB - Let $\lbrace X(t),\; t\in [0,1]^n\rbrace$ be a stochastically continuous, separable, Gaussian process with $E[X(t+h)-X(t)]^2=\sigma ^2(\vert h\vert )$. A sufficient condition, in terms of the monotone rearrangement of $\sigma$, 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.
LA - eng
UR - http://eudml.org/doc/74167
ER -

## References

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1. [1] R. P. BOAS Jr. and M. B. MARCUS, Inequalities involving a function and its inverse, SIAM J. Math. Anal., 4 (1973). Zbl0235.26009MR48 #8724
2. [2] DUDLEY R. M., The sizes of compact subsets of Hilbert space and continuity of Gaussian processes, J. Functional Analysis, 1 (1967), 290-330. Zbl0188.20502MR36 #3405
3. [3] R. M. DUDLEY, Sample functions of the Gaussian process, Ann. of Probability, 1 (1973), 66-103. Zbl0261.60033MR49 #11605
4. [4] X. FERNIQUE, Continuité des processus Gaussiens, C.R. Acad. Sci. Paris, 258 (1964), 6058-6060. Zbl0129.30101MR29 #1662
5. [5] G. H. HARDY, J. E. LITTLEWOOD and G. POLYA, Inequalities, Cambridge Univ. Press, (1934), Cambridge, England. Zbl0010.10703JFM60.0169.01
6. [6] N. C. JAIN, Conditions for the continuity of sample paths of a Gaussian process, unpublished manuscript, (1972).
7. [7] N. C. JAIN and G. KALLIANPUR, A note on the uniform convergence of stochastic processes, 41 (1970), 1360-1362. Zbl0232.60040MR42 #6931
8. [8] J. P. KAHANE, Some random series of functions, (1968), D. C. Heath, Lexington, Mass. Zbl0192.53801MR40 #8095
9. [9] E. LUKACS, Characteristic functions, Second Edition, (1970), Hafner, New York. Zbl0201.20404MR49 #11595
10. [10] M. B. MARCUS, A comparaison of continuity conditions for Gaussian processes., Ann. of Probability, 1 (1973), 123-130. Zbl0265.60039MR49 #11606
11. [11] M. B. MARCUS, Continuity of Gaussian processes and random Fourier series, Ann. of Probability, 1 (1973), 968-981. Zbl0277.60022MR50 #8673
12. [12] M. B. MARCUS and L. A. SHEPP, Continuity of Gaussian processes., Trans. Amer. Math. Soc., 151 (1970), 377-392. Zbl0209.49201MR41 #9340
13. [13] J. L. DOOB, Stochastic processes, (1953), John Wiley and Sons, New York. Zbl0053.26802MR15,445b

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