Some results on the continuity of stable processes and the domain of attraction of continuous stable processes

We study the continuity of p-stable stochastic processes (1 __ p 2) and their domain of attraction on the Banach space C. We can improve several recent results by weakening the assumptions of the metric entropy conditions. We also give some necessary conditions for the continuity of p-stable random Fourier integrals which extend results of Nisio and Salem-Zygmund for the case p = 2. RESUME. Nous étudions la continuité des processus p-stables (1 __ p 2) et leur domaine d’attraction sur l’espace de Banach C. Notre étude permet d’affaiblir les hypotheses sur 1’ « entropie métrique » dans plusieurs travaux récents. Nous donnons aussi des conditions necessaires pour la continuité des « intégrales de Fourier aléatoires » qui étendent au cas p-stable des résultats de Nisio et Salem-Zygmund. In [8] ] necessary and sufficient conditions are obtained for the continuity of strongly stationary p-stable random Fourier series. Methods used in [8 ] enable us to improve upon some of the results in [1 ] [3 and [4] relating to the continuity of p-stable stochastic processes and the characterization of their domains of attraction by weakening the requirements Annales de l’Institut Henri Poincaré Probabilités et Statistiques Vol. 20, 0246-0203 84/02/177/23/$ 4,30/(e) Gauthier-Villars 178 M. B. MARCUS AND G. PISIER on the size of the metric entropy used in these results. We do this in Section 1. In Section 2 we give some necessary conditions for the continuity of p-stable random Fourier series and integrals, 1 _ p 2, that are analogous to the classical results of Salem and Zygmund for series in the case p = 2. 1 SUFFICIENT CONDITIONS FOR CONTINUITY AND THE CENTRAL LIMIT THEOREM Let D: R + -~ R+ be an increasing convex function with O(0) = 0. For any probability space (Q, ~ , P) we denote by L~’(d P) the so called « Orlicz space » formed by all measurable functions f : S~ -~ C for which there is a c > 0 such that We equip this space with the norm , We define and We will consider the Orlicz spaces 2 q o~. We will also be concerned with the weak Lp,co spaces defined as follows: These are the spaces of all real valued random variables for which P( I X > ~.) = 0(1/~,p), 1 p 2. For these spaces we consider the function which is equivalent to a norm for p > 1. Let (T, p) be a compact metric or pseudo-metric space. We define by N(T, p ; E) the minimum number of open balls of radius s > 0 in the p metric or pseudo-metric, with centers in T, that is necessary to cover T. We define Annales de l’lnstitut Henri Poincaré Probabilités et Statistiques 179 SOME RESULTS ON THE CONTINUITY OF STABLE PROCESSES and Jq( p ; oo ) = Jq( p), 2 q ~. The first lemma is a variant of Dudley’s Theorem for Gaussian processes. It has been observed in various forms by many authors. LEMMA 1.1. Let {X(t), be in 2 q oo and satisfy If Jq(p) oo, 2 q oo then { X(t ), t E T } has a version with continuous sample paths and where CPq(u) = u (log + log 1/u) 1 ~q, 2 q = u (log + log + log 1/u), p = sup p(s, t ) and Dq is a constant depending only on q. Furthermore, s,tET for any to eT where D~ is a constant depending only on q. Proof This is proved in Chapter II, Theorem 3.1 [7], in the case q = 2. As we comment in Lemma 3 . 2 [8 ], the proof for 2 q oo is completely similar since ( 1.1 ) implies Likewise, the proof when follows because in this case ( 1.1 ) implies The next Theorem gives conditions which imply tightness for normed sums of i. i. d. C(T) valued random variables, where C(T) denotes the Banach space of continuous functions on T with the sup norm. THEOREM 1. 2. Let {X(t), a real valued stochastic process with continuous sample paths. If p > 1, we assume that E 00 and EX(t ) = 0 for all let {X(t), be symmetric. Let be i. i. d. copies Let r be a continuous metric or pseudo-metric on T and define