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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

MSC:
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
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