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Characterizations of almost surely continuousp-stable random Fourier series and strongly stationary processes

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References

  1. Badrikian, A. &Chevet, S.,Mesures cylindriques, espaces de Wiener et fonctions aléatoires gaussiennes. Lecture Notes in Math. 379 (1974). Springer-Verlag, New York.

    MATH  Google Scholar 

  2. de Acosta, A., Stable measures and semi-norms.Ann. Probab., 3 (1975), 365–875.

    Google Scholar 

  3. Bergh, J. &Löfström, J.,Interpolation spaces. Springer-Verlag, New York (1976).

    MATH  Google Scholar 

  4. Breiman, L.,Probability. Addison-Wesley, Reading, Mass. (1968).

    MATH  Google Scholar 

  5. Bretagnolle, J., Dacunha-Castelle, D. &Krivine, J. L., Lois stable et espacesL p.Ann. Inst. H. Poincaré Sect. B, 2 (1966), 231–259.

    MATH  MathSciNet  Google Scholar 

  6. Dudley, R. M., The sizes of compact subsets of Hilbert space and continuity of Gaussian processes.J. Funct. Anal., 1 (1967), 290–330.

    Article  MATH  MathSciNet  Google Scholar 

  7. —, Sample functions of the Gaussian process,Ann. Probab., 1 (1973), 66–103.

    MATH  MathSciNet  Google Scholar 

  8. Erhard, A. &Fernique, X., Fonctions aléatoires stable irrégulières.C.R. Acad. Sci. Paris Math. Sér A, 292 (1981), 999–1001.

    Google Scholar 

  9. Feller, W.,An introduction to probability theory and its applications, Vol. II. First edition (1966), J. Wiley & Sons, New York.

    MATH  Google Scholar 

  10. Fernique, X., Régularité des trajectoires des fonctions aléatoires gausiennes.Lecture Notes in Math., 480 (1975), 1–96.

    Article  MATH  MathSciNet  Google Scholar 

  11. —, Continuité et théorème central limite pour les transformées de Fourier des mesures aléatoires du second ordre.Z. Wahrsch. Verw. Gebiete, 42 (1978), 57–66.

    Article  MATH  MathSciNet  Google Scholar 

  12. Fernique, X., Régularité de fonctions aléatoires non gaussiennes.Ecole d’Eté de St. Flour, (1981). Lecture Notes in Math. no 976 (1983), 1–74.

  13. Hoffman-Jørgensen, J., Sums of independent Banach space valued random variables.Studia Math., 52 (1974), 159–186.

    MathSciNet  Google Scholar 

  14. Jain, N. &Marcus, M. B., Integrability of infinite sums of independent vector valued random variables.Trans. Amer. Math. Soc., 212 (1975), 1–36.

    Article  MATH  MathSciNet  Google Scholar 

  15. —, Continuity of subgaussian processes, inAdvances in Probability, Vol. 4 (1978). Edited by J. Kuelbs, M. Dekker, New York.

    Google Scholar 

  16. Kahane, J. P.,Some random series of functions. D. C. Heath, Lexington, Mass. (1968).

    MATH  Google Scholar 

  17. LePage, R., Woodroofe, M. &Zinn, J., Convergence to a stable distribution via order statistics.Ann. Probab., 9 (1981), 624–632.

    MATH  MathSciNet  Google Scholar 

  18. LePage, R.,Multidimensional infinitely divisible variables and processes, Part I: Stable case. Technical report 292, Stanford University.

  19. Marcus, M. B., Continuity of Gaussian processes and random Fourier series.Ann. Probab., 1 (1973), 968–981.

    MATH  Google Scholar 

  20. —, Continuity and central limit theorem for random trigonometric series.Z. Wahrsch. Verw. Gebiete, 42 (1978), 35–56.

    Article  MATH  MathSciNet  Google Scholar 

  21. Marcus, M. B. &Pisier, G.,Random Fourier series with applications to harmonic analysis. Ann. Math. Studies, Vol. 101 (1981). Princeton Univ. Press, Princeton, N.J.

    MATH  Google Scholar 

  22. Nanopoulis, C. & Nobelis, P.,Etude de la régularité des fonctions aléatoires et de leur propriétés limites. These de 3e cycle, Strasbourg (1977).

  23. Neveu, J.,Discrete-parameter martingales. North Holland, New York (1975).

    MATH  Google Scholar 

  24. Pisier, G., Sur l’espace de Banach des séries de Fourier aléatoires presque sûrement continues.Séminaire sur la géométrie des espaces de Banach 77–78. Ecole Polytechnique, Palaiseau.

  25. —, De nouvelles caractérisations des ensembles de Sidon. Mathematical Analysis and Applications.Adv. in Math. Suppl. Stud., 7B (1981), 686–725.

    Google Scholar 

  26. Pisier, G., Some applications of the metric entropy condition to harmonic analysis, inBanach spaces, Harmonic Analysis and Probability, Proceedings 80–81. Springer Lecture Notes, 995 (1983), 123–154.

  27. Rényi, A., Problems in ordered samples.Selected Translations in Math. Stat. and Prob., 13 (1973), 289–298.

    MATH  Google Scholar 

  28. Rodin, V. A. &Semyonov, E. M., Rademacher series in symmetric spaces.Anal. Math., 1 (1975), 207–222.

    Article  MATH  MathSciNet  Google Scholar 

  29. Weber, M., Analyse infinitésimale de fonctions aléatoires.Ecole d’Eté de St. Flour (1981). Lecture Notes in Math. no 976 (1983), 383–465.

  30. Wells, J. H. &Williams, L. R.,Embeddings and extensions in analysis. Ergebnisse Band 84, Springer-Verlag, New York (1975).

    MATH  Google Scholar 

  31. van Zuijlen, M. C. A., Properties of the empirical distribution function for independent nonidentically distributed random variables.Ann. Probab., 6 (1978), 250–266.

    MATH  Google Scholar 

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This research was supported in part by a grant from the National Science Foundation.

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Marcus, M.B., Pisier, G. Characterizations of almost surely continuousp-stable random Fourier series and strongly stationary processes. Acta Math 152, 245–301 (1984). https://doi.org/10.1007/BF02392199

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