Abstract
LetB be a separable Banach space and let {ξ:|ξ|≤1} denote the unit ball ofB *. LetX be a symmetricp-stableB-valued random variable and let {X j } n j=1 be i.i.d. copies ofX. LetB 1 be a finite-dimensional Banach space with a symmetric unconditional basis {y j } n j=1 . An upper bound is obtained for\(E \sup _{|\xi | \leqslant 1} \parallel \sum\nolimits_{j = 1}^n {\xi (X_j )} y_j \parallel \) that improves the one given by Giné, Marcus and Zinn [J. Functional Anal. 63, 47–73 (1985)].
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Chevet, S. (1978) Séries de variables aléatoire gaussiennes a valeurs dansE \(\hat \oplus _\varepsilon \) F. Applications aux produits d'espaces de Wiener abstraits,Seminaire sur la géometrie des espaces de Banach 1977–78, Exp XIX, Ecole Polytechnique, Paris.
Chevet, S. (1978). Quelques nouveaux résultats sur les mésures cylandrique. InLecture Notes in Mathematics, Vol. 644, pp. 125–158, Springer-Verlag, Berlin.
Carmona, R. (1978). Tensor product of Gaussian measures, inLecture Notes in Mathématics, Vol. 644, Springer-Verlag, Berlin, pp. 96–124.
Giné, E., Marcus, M. B. and Zinn, J. (1985). A version of Chevet's theorem for stable processes.J. Functional Anal. 63, 47–73.
Araujo, A., and Giné, E. (1980)The Central Limit Theorem for Real and Banach Valued Random Variables, Wiley, New York.
Marcus, M. B., and Pisier, G. (1984). Characterisation of almost surely continuousp-stable random Fourier series and strongly stationary processes.Acta Math. 152, 245–301.
Fernique, X. (1975). Regularité des trajectoires des fonctions aleatoire gaussiennes. InLecture Notes in Mathematics, Vol. 480, Springer-Verlag, Berlin, pp. 1–96.
Ledoux, M., and Talagrand, M. (1986). Limit theorems for empirical processes, preprint.
Jain, N., and Marcus, M. B. (1975). Integrability of infinite sums of independent vectorvalued random variables.Trans. Am. Math. Soc. 212, 1–36.
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Marcus, M.B., Talagrand, M. Chevet's theorem for stable processes II. J Theor Probab 1, 65–92 (1988). https://doi.org/10.1007/BF01076288
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DOI: https://doi.org/10.1007/BF01076288