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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Integrability of infinite sums of independent vector-valued random variables

Authors: Naresh C. Jain and Michael B. Marcus
Journal: Trans. Amer. Math. Soc. 212 (1975), 1-36
MSC: Primary 60G15; Secondary 60B05, 60G50
MathSciNet review: 0385995
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Abstract: Let B be a normed vector space (possibly a Banach space, but it could be more general) and $ \{ {X_n}\} $ a sequence of B-valued independent random variables on some probability space. Let $ {S_n} = \Sigma _{j = 1}^n{X_j},M = {\sup _n}\vert{S_n}\vert$ and $ S = {\lim _n}{S_n}$ is norm, whenever it exists. Assuming that S exists or $ M < \infty $ a.s. and given certain nondecreasing functions $ \varphi $, we find conditions in terms of the distributions of $ \left\Vert {{X_n}} \right\Vert$ such that $ E(\varphi (M))$ or $ E(\varphi (\left\Vert S \right\Vert))$ is finite.

Let $ \{ {u_n}\} $ be a sequence of elements in B and $ \{ {\varepsilon _n}\} $ a sequence of independent, identically distributed random variables such that $ P\{ {\varepsilon _1} = 1\} = P\{ {\varepsilon _1} = - 1\} = {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}$. We prove some comparison theorems which generalize the following contraction principle of Kahane: If $ \{ {\lambda _n}\} $ is a bounded sequence of scalars, then $ \Sigma {\varepsilon _n}{u_n}$ converges in norm a.s. (or is bounded a.s.) implies the corresponding conclusion for the series $ \Sigma {\lambda _n}{\varepsilon _n}{u_n}$. Some generalizations of this contraction principle have already been carried out by Hoffmann-Jørgensen. All these earlier results are subsumed by ours.

Applications of our results are made to Gaussian processes, random Fourier series and other random series of functions.

References [Enhancements On Off] (What's this?)

  • [1] W. Feller (1966), An introduction to probability theory and its applications. Vol. II, Wiley, New York. MR 35 #1048. MR 0210154 (35:1048)
  • [2] X. Fernique (1970), Intégrabilité des vecteurs Gaussiens, C. R. Acad. Sci. Paris Sér. A-B 270, A1698-A1699. MR 42 #1170. MR 0266263 (42:1170)
  • [3] J. Hoffmann-Jørgensen (1973), Sums of independent Banach space valued random variables, Aarhus Universitet Matematisk Institut, Denmark (preprint).
  • [3a] -(1974), Sums of independent Banach space valued random variables, Studia Math. 52 (1974), 159-186. MR 0356155 (50:8626)
  • [4] N. C. Jain and M. B. Marcus (1974), Sufficient conditions for the continuity of stationary Gaussian processes and applications to random series of functions, Ann. Inst. Fourier (Grenoble) 24, 117-141. MR 0413239 (54:1356)
  • [5] J.-P. Kahane (1968), Some random series of functions, Heath, Lexington, Mass. MR 40 #8095. MR 0254888 (40:8095)
  • [6] H. J. Landau and L. A. Shepp (1970), On the supremum of a Gaussian process, Sankhyā Ser. A 32, 369-378. MR 44 #3381. MR 0286167 (44:3381)
  • [7] M. B. Marcus (1974), Uniform convergence of random Fourier series, Ark. Mat. (to appear). MR 0372994 (51:9196)
  • [8] G. Pisier (1973), Type des espaces normes, C. R. Acad. Sci. Paris Sér. A--B 276, A1673-A1676. MR 0342989 (49:7733)
  • [9] A. V. Skorohod (1970), A note on Gaussian measures in Banach space, Teor. Verojatnost. i Primenen. 15, 519-520. (Russian) MR 43 #3417. MR 0277684 (43:3417)

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Keywords: Integrability, infinite series, Banach space valued random variables, convergence in norm, contraction principle, uniformly nondegenerate, Rademacher sequence
Article copyright: © Copyright 1975 American Mathematical Society

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