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Zbl 0536.60048
Marcus, M.B.; Pisier, G.
Some results on the continuity of stable processes and the domain of attraction of continuous stable processes. (English)
[J] Ann. Inst. Henri Poincaré, Probab. Stat. 20, 177-199 (1984). ISSN 0246-0203

Direct embedding into Gaussian or Rademacher processes and use of the properties of these processes had allowed {\it M. Marcus} and the reviewer [e.g., Aspects statistiques et aspects physique des processus gaussiens, Saint-Flour 1980, Colloq Int. C.N.R.S. 307, 361-383 (1981; Zbl 0516.60011), and "Some results on the domain of attraction of stable measures on C(K)." Probab. Math. Stat. 2, 125-147 (1982)] and {\it A. Araujo} and the reviewer [The central limit theorem for real and Banach valued random variables. Wiley (1980; Zbl 0457.60001), ch. 7] to obtain sufficient conditions for the existence of p-stable processes with continuous sample paths and for domains of attraction. \par The techniques in the important work of the present authors, "Characterization of almost surely continuous p-stable random Fourier series and strongly stationary processes", Acta Math., to appear, which are sharp and yield best results in the case of strong stationarity are applied in this paper to obtain the best metric entropy sufficient conditions for the domain of attraction problem and for sample continuity of p-stable processes, $1\le p<2$ (thus improving on the previous ones). Necessary conditions for the continuity of p-stable random Fourier integrals which extend results of {\it M. Nisio} [Nagoya Math. J. 34, 89- 104 (1969; Zbl 0174.493)] and Salem and Zygmund for the case $p=2$ are also obtained.
[E.Giné]

MSC 2000:: *60G17 Sample path properties
60B12 Limit theorems for vector-valued random variables (inf.-dim.case)
60B11 Probability theory on linear topological spaces
60B15 Probability measures on groups

Keywords: sample continuous stable processes; domains of attraction; random Fourier integrals; metric entropy

Citations: Zbl 0516.60011; Zbl 0457.60001; Zbl 0174.493

Cited in: Zbl 0709.60008

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