# 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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