A functional limit theorem for Erdös and Rényi's law of large numbers
Abstract
We use the theory of large deviations on function spaces to extend Erdös and Rényi's law of large numbers. In particular, we show that with probability 1, the double-indexed set of paths {W } defined by {Mathematical expression} where {Mathematical expression}, {X : ≧1} is an iid sequence of random variables, and h(N)=[clog N] is relatively compact; the limit set is given by the set [x:I (x)≦1/c] where I (x) = ∫ I(x′(t))dt and I is Cramér's rate function. © 1994 Springer-Verlag. N, n i i 0 * * 1
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