ON DEVIATIONS IN SKOROKHOD-STRASSEN APPROXIMATION SCHEME

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 {Tii>0} such that {Mathematical expression} are i. i. d. with d. f. F. Let {Mathematical expression} Strassen showed Z=O(1) wp 1. We prove Z=(2 Β)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 {Mathematical expression} is not a sum of independent random variables. © 1969 Springer-Verlag.