, Volume 88, Issue 3, pp 381-404

Nearby variables with nearby conditional laws and a strong approximation theorem for Hilbert space valued martingales

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In this paper we focus on sequences of random vectors which do not admit a strong approximation of their partial sums by sums of independent random vectors. In the first part we prove conditional versions of the Strassen-Dudley theorem. We apply these in the second part of the paper to obtain strong invariance principles for vector-valued martingales which, when properly normalized, converge in law to a mixture of Gaussian distributions.

Research partially supported by the Air Force Office of Scientific Research Contract NO. F49260 85 C 0144
Part of this work was done while the second author was visiting the Department of Statistics and the Center for Stochastic Processes, University of North Carolina, Chapel Hill. He thanks the members of the Department of Statistics for their hospitality