Summary
In deriving his strong invariance principles, Strassen used a construction of Skorokhod: if the univariate d. f. F has first, second, and fourth moments 0, 1, and Β<t8, respectively, then there is a probability space on which are defined a standard Brownian motion {ξ(t), t≧0} and a sequence of nonnegative i.i.d. Skorokhod random variables {T i i>0} such that
are i. i. d. with d. f. F. Let
Strassen showed Z=O(1) wp 1. We prove Z=(2Β)1/4 wp 1. Consequently Z=0 wp 1 implies F is Gaussian, answering a special case of a question of Strassen. Analogous results hold for cases where \(\xi \left( {n\sum\limits_1^n {T_i } } \right)\) is not a sum of independent random variables.
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Kiefer, J. On the deviations in the Skorokhod-Strassen approximation scheme. Z. Wahrscheinlichkeitstheorie verw Gebiete 13, 321–332 (1969). https://doi.org/10.1007/BF00539208
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DOI: https://doi.org/10.1007/BF00539208