Let $X_1,X_2, . . .$ be an i.i.d. sequence of random variables with
$E[X_1] = 1/2$ and $\operatorname{Var}[X_i] = 2$.
I want to Compute:
$$P\left(\lim_{n\to\infty} (X_1+X_2+…+X_n)/n > 1\right)$$
My try:
$P(\lim_{n->\infty} (X1+X2+…)/n > 1) = 1-P(\lim_{n->\infty} (X1+X2+…)/n \leq 1) = 1-P(\lim_{n->\infty} ((X1+X2+…)/n-1/2)\sqrt{2/n} \leq (1/2)/\sqrt{2/n})$
as I know that: $((X1+X2+…)/n-1/2)\sqrt{2/n}~N(0,1)$ From Central limit theorem. But my problem is the limit inside.
My Questions:
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How can I continue from here
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Is there a way to solve this exactly without using Central limit theorem?
Best Answer
From Strong Law of Large numbers:
$$P\left(\lim_{n\to \infty} \frac{\sum_{i=1}^nX_i}{n}=1/2\right)=1 \implies P\left(\lim_{n\to \infty} \frac{\sum_{i=1}^nX_i}{n}\ne1/2\right) = 0.$$
Since $1/2\neq 1$, we have that $P\left(\lim_{n\to \infty} \frac{\sum_{i=1}X_i}{n}>1\right)=0$.