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Miklós Schweitzer 1986, Problem 10

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September 12, 2016
Miklos Schweitzercollege contestsprobabilityrandom variables

Problem Statement

Let X1,X2X_1, X_2 be independent, identically distributed random variables such that Xi≥0X_i\geq 0 for all ii. Let EXi=m\mathrm EX_i=m, Var(Xi)=σ2<∞\mathrm{Var} (X_i)=\sigma ^2<\infty. Show that, for all 0<α≤10<\alpha\leq 1 lim⁡n→∞n Var([X1+…+Xnn]α)=α2σ2m2(1−α)\lim_{n\to\infty} n\,\mathrm{Var} \left( \left[ \frac{X_1+\ldots +X_n}{n}\right] ^\alpha\right)=\frac{\alpha ^ 2 \sigma ^ 2}{m^{2(1-\alpha)}} [Gy. Michaletzki]
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