Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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:


share|cite|improve this question
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):

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

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.