Iterating Random Functions on a Finite Set
نویسندگان
چکیده
Choose random functions f1, f2, f3, . . . independently and uniformly from among the nn functions from [n] into [n]. For t > 1, let gt = ft ◦ ft−1 ◦ · · · ◦ f1 be the composition of the first t functions, and let T be the smallest t for which gt is constant(i.e. gt(i) = gt(j) for all i, j). The goal of this paper is to determine the asymptotic distribution of T . We prove that, for any positive real number x, lim n→∞ Pr( T n ≤ x) = x ∫ 0 f(y)dy, where f(y) = ∑ k≥2 (−1)ke−y( k 2 )(2k − 1) (
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