Chapter 4: E4.7 Solution

Working problems is a crucial part of learning mathematics. No one can learn topology merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.

James R. Munkres

What’s fair about a fair game?

Let $X_{1},X_{2},\ldots$ be independent RVs such that $X_{n}=\begin{cases} n^{2}-1 & \text{with probability }n^{-2}\\ -1 & \text{with probability }1-n^{-2}. \end{cases}$ Prove that $\mathbb{E}(X_{n})=0$ , $\forall n$ , but that if $S_{n}=X_{1}+X_{2}+\dotsb+X_{n}$ , then $\tfrac{S_{n}}{n}\to-1,\text{ a.s.}$

$\mathbb{E}(X_{n})=\tfrac{n^{2}-1}{n^{2}}-(1-\tfrac{1}{n^{2}})=0$ , and $\sum_{n\in\mathbb{N}}\mathbb{P}(X_{n}\ge0)<\infty$ , so that $\mathbb{P}(X_{n}\ge0,\text{ i.o.})=0$ , and $\mathbb{P}(X_{n}=-1,\text{ ev})=1$ , implying $\mathbb{P}(\tfrac{S_{n}}{n}\to-1)=1$ .

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Chapter 4: E4.7 Solution

Chapter 4: E4.7 Solution