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Miklós Schweitzer 1954- Problem 5

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September 29, 2015
probabilitycollege contests

Problem Statement

5. Let ξ1,ξ2,…,ξn,...\xi _{1},\xi _{2},\dots ,\xi _{n},... be independent random variables of uniform distribution in (0,1)(0,1). Show that the distribution of the random variable
ηn=n∏k=1n(1−ξkk)(n=1,2,...)\eta _{n}= \sqrt[]{n}\prod_{k=1}^{n}(1-\frac{\xi _{k}}{k}) (n= 1,2,...)
tends to a limit distribution for n→∞n \to \infty . (P. 6)
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