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Transactions of the American Mathematical Society

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

 

 

Almost sure invariance principles for sums of $ B$-valued random variables with applications to random Fourier series and the empirical characteristic process


Authors: Michael B. Marcus and Walter Philipp
Journal: Trans. Amer. Math. Soc. 269 (1982), 67-90
MSC: Primary 60F17; Secondary 60B12
MathSciNet review: 637029
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Abstract: We establish an almost sure approximation of the partial sums of independent, identically distributed random variables with values in a separable Banach space $ B$ by a suitable $ B$-valued Brownian motion under the hypothesis that the partial sums can be $ {L^1}$-closely approximated by finite-dimensional random variables. We show that this hypothesis is satisfied if the given random variables are random Fourier series or related stochastic processes. As an application we obtain an almost sure approximation of the empirical characteristic process by a suitable $ {\mathbf{C}}(K)$-valued Brownian motion whenever the empirical characteristic process satisfies the central limit theorem.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1982-0637029-8
Keywords: Invariance principles, Banach space valued random variables, Brownian motion, random Fourier series, empirical characteristic process, central limit theorem
Article copyright: © Copyright 1982 American Mathematical Society