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Let $X_1,X_2,...$ independent identically distributed (i.i.d.) random variables with $E(X_i)=2$ and $Var(X_i)=1$. Find the almost sure limit of:

$$Y_n={{\sum_{i=1}^nX_i\sum_{i=1}^n(X_i-2)^2}\over{n^2}}$$

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up vote 1 down vote accepted

Write $Y_n$ as a product:

$$Y_n = \frac{\sum_{i=1}^n X_i}{n} \cdot \frac{\sum_{i=1}^n (X_i-2)^2}{n}$$

Now you can apply the Strong Law of Large Numbers (since the random variables are iid):

$$\frac{\sum_{i=1}^n X_i}{n} \to \mathbb{E}X_i = 2 \quad \text{a.s.} \qquad (n \to \infty) \\ \frac{\sum_{i=1}^n (X_i-2)^2}{n} = \frac{\sum_{i=1}^n (X_i-\mathbb{E}(X_i))^2}{n} \to \mathbb{E}((X_i-\mathbb{E}(X_i))^2) = \text{Var}(X_i)=1 \quad \text{a.s.} \quad (n \to \infty)$$

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