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11

Part of 1970 Miklós Schweitzer

Problems(1)

Miklos Schweitzer 1970_11

Source:

10/22/2008
Let ξ1,ξ2,... \xi_1,\xi_2,... be independent random variables such that Eξn=m>0 E\xi_n=m>0 and Var(ξn)=σ2<∞  (n=1,2,...) . \textrm{Var}(\xi_n)=\sigma^2 < \infty \;(n=1,2,...)\ . Let {an} \{a_n \} be a sequence of positive numbers such that an→0 a_n\rightarrow 0 and ∑n=1∞an=∞ \sum_{n=1}^{\infty} a_n= \infty. Prove that P(lim⁡n→∞∑k=1nakξk=∞)=1. P \left( \lim_{n\rightarrow \infty} %Error. "diaplaymath" is a bad command. \sum_{k=1}^n a_k \xi_k =\infty \right)=1. P. Revesz
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