zbMATH — the first resource for mathematics

Characterizations of almost surely continuous p-stable random Fourier series and strongly stationary processes. (English) Zbl 0547.60047
In this important paper the authors extend the Dudley-Fernique ”metric entropy” characterization of the a.s. continuity of sample paths of stationary Gaussian processes to strongly stationary p-stable processes, \(1<p<2\), indexed by a compact neighbourhood of the unit element of a locally compact Abelian group G. In contrast with the Gaussian case \(p=2\) the class of p-stable processes with harmonic representation forms only a subclass of all stationary p-stable processes. Therefore a stochastic process X(t) on G is called here strongly stationary p-stable, \(0<p\leq 2\), if there exists a finite positive Radon measure m on the dual group \(\Gamma\) such that for all \(t_ 1,...,t_ n\in G\) and complex numbers \(a_ 1,...,a_ n E \exp iRe\sum^{n}_{j=1}\bar a_ jX(t_ j)=\exp -\int_{\Gamma}|\sum^{n}_{j=1}\bar a_ j\gamma (t_ j)|^ pdm(\gamma).\)In particular, if m is a discrete measure then X(t) is a random Fourier series. Consequently, the above result enables the authors to obtain necessary and sufficient conditions for the a.s. continuity of random Fourier series \(\sum_{\gamma}a_{\gamma}\xi_{\gamma}\gamma (t)\) with independent coefficients \(\xi_{\gamma}\) which do not have finite second moment. For \(\sup_{\gamma}E|\xi_{\gamma}|^ 2<\infty\) and \(\inf_{\gamma}E|\xi_{\gamma}| >0\) a necessary and sufficient condition was given earlier by the authors [Random Fourier series with application to harmonic analysis. (1981; Zbl 0474.43004)]. It has been shown by A. Ehrhard and X. Fernique [C. R. Acad. Sci., Paris, Sér. I 292, 999-1001 (1981; Zbl 0472.60038)] that in general Slepian’s lemma can not be extended directly from Gaussian to p- stable processes. However for strongly stationary p-stable processes the authors deduce a version of Slepian’s lemma as well as interesting generalizations by comparing general p-stable processes with Gaussian processes.
The key idea in the proofs of main results is to employ a very useful representation of any (strongly stationary) p-stable process as a mixture of (stationary) Gaussian processes obtained first by R. LePage [Tech. Rep. 292, Stanford Univ. (1980); cf. also R. LePage, M. Woodroofe and J. Zinn, Ann. Probab. 9, 624-632 (1981; Zbl 0465.60031)].
The paper contains also some applications, of independent interest, to \(L^ p\) space theory and harmonic analysis. For example, it is proved that a contraction from a finite subset of \(L^ p\) into a Hilbert space H has an extension with a relatively small norm to a mapping from \(L^ p\) to H. Following the previous works of the second author [De nouvelles caractérisations des ensembles de Sidon. Adv. Math., Suppl. Stud. 7B, 686-725 (1981); Banach spaces, harmonic analysis and probability theory, Proc. Spec. Year Analysis, Univ. Conn. 1980-81, Lect. Notes Math. 995, 123-154 (1983; Zbl 0517.60043)] it is shown that for \(1<p<2\) the space of all p-stable a.s. continuous random Fourier series can be identified with the predual of a certain space of multipliers.
Reviewer: A.Weron

60G17 Sample path properties
60G10 Stationary stochastic processes
60E07 Infinitely divisible distributions; stable distributions
46E99 Linear function spaces and their duals
43A25 Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
Full Text: DOI
[1] Badrikian, A. &Chevet, S.,Mesures cylindriques, espaces de Wiener et fonctions aléatoires gaussiennes. Lecture Notes in Math. 379 (1974). Springer-Verlag, New York. · Zbl 0288.60009
[2] de Acosta, A., Stable measures and semi-norms.Ann. Probab., 3 (1975), 365–875. · Zbl 0303.60041
[3] Bergh, J. &Löfström, J.,Interpolation spaces. Springer-Verlag, New York (1976). · Zbl 0344.46071
[4] Breiman, L.,Probability. Addison-Wesley, Reading, Mass. (1968).
[5] Bretagnolle, J., Dacunha-Castelle, D. &Krivine, J. L., Lois stable et espacesL p .Ann. Inst. H. Poincaré Sect. B, 2 (1966), 231–259. · Zbl 0139.33501
[6] Dudley, R. M., The sizes of compact subsets of Hilbert space and continuity of Gaussian processes.J. Funct. Anal., 1 (1967), 290–330. · Zbl 0188.20502
[7] –, Sample functions of the Gaussian process,Ann. Probab., 1 (1973), 66–103. · Zbl 0261.60033
[8] Erhard, A. &Fernique, X., Fonctions aléatoires stable irrégulières.C.R. Acad. Sci. Paris Math. Sér A, 292 (1981), 999–1001. · Zbl 0472.60038
[9] Feller, W.,An introduction to probability theory and its applications, Vol. II. First edition (1966), J. Wiley & Sons, New York. · Zbl 0138.10207
[10] Fernique, X., Régularité des trajectoires des fonctions aléatoires gausiennes.Lecture Notes in Math., 480 (1975), 1–96.
[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. · Zbl 0393.60020
[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. · Zbl 0265.60005
[14] Jain, N. &Marcus, M. B., Integrability of infinite sums of independent vector valued random variables.Trans. Amer. Math. Soc., 212 (1975), 1–36. · Zbl 0318.60036
[15] –, Continuity of subgaussian processes, inAdvances in Probability, Vol. 4 (1978). Edited by J. Kuelbs, M. Dekker, New York.
[16] Kahane, J. P.,Some random series of functions. D. C. Heath, Lexington, Mass. (1968). · Zbl 0192.53801
[17] LePage, R., Woodroofe, M. &Zinn, J., Convergence to a stable distribution via order statistics.Ann. Probab., 9 (1981), 624–632. · Zbl 0465.60031
[18] LePage, R.,Multidimensional infinitely divisible variables and processes, Part I: Stable case. Technical report 292, Stanford University. · Zbl 0469.60012
[19] Marcus, M. B., Continuity of Gaussian processes and random Fourier series.Ann. Probab., 1 (1973), 968–981. · Zbl 0277.60022
[20] –, Continuity and central limit theorem for random trigonometric series.Z. Wahrsch. Verw. Gebiete, 42 (1978), 35–56. · Zbl 0382.60041
[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. · Zbl 0474.43004
[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). · Zbl 0345.60026
[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.
[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.
[28] Rodin, V. A. &Semyonov, E. M., Rademacher series in symmetric spaces.Anal. Math., 1 (1975), 207–222. · Zbl 0315.46031
[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). · Zbl 0324.46034
[31] van Zuijlen, M. C. A., Properties of the empirical distribution function for independent nonidentically distributed random variables.Ann. Probab., 6 (1978), 250–266. · Zbl 0396.60040
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.