- #1

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Let X

_{i}be random variables identically distributed, with mean 0, such that the cumulative distribution is = 0 for all -1 < x < 1. So, I believe it is clear that for all n, the cumulative distribution of Z = (X

_{1}+ X

_{2}... X

_{n})/n is = 0 for all x < -1. But the central limit theorem implies that this distribution (normalized by the square root of n), converges in distribution to the Normal distribution. So, for some n sufficiently large, the cumulative distribution of Z must be > 0 for some x < -1. Where is the fallacy in this paradox?

thx.