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

Naresh C. Jain; Michael B. Marcus

Annales de l'institut Fourier (1974)

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

Abstract

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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.

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

Citations in EuDML Documents

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  1. Michael B. Marcus, Gilles Pisier, Random Fourier series on locally compact abelian groups
  2. B. Heinkel, Un théorème de la limite centrale dans C ( S )
  3. G. Pisier, Sur l'espace de Banach des séries de Fourier aléatoires presque sûrement continues
  4. Thierry Jeulin, Marc Yor, Moyennes mobiles et semimartingales
  5. G. Pisier, Conditions d'entropie assurant la continuité de certains processus et applications à l'analyse harmonique
  6. Constantin Nanopoulos, Photis Nobelis, Régularité et propriétés limites des fonctions aléatoires

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