Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I'm reading Probability Theory, and ran into the following exercise:

Prove that if $X_n \in L^2$ are uncorrelated and identically distributed random variables, then $$\frac{1}{n} \sum_{k=1}^n X_k \to \mathbb{E}[X_1]\text{ almost surely.}$$

This seems a bit weird, since it is given as a theorem that the convergence happens in probability. (Which implies that I can find a subsequence that converges almost surely.) They don't mention this strengthening of the theorem in the text, so I'm wondering if what the exercise claims is true.

share|improve this question
    
When you say "uncorrelated" do you mean "independent"? (To me "uncorrelated" could mean $\text{Cov}(X_i, X_j) = 0, i \neq j$ which is a weaker statement than "independent" in general.) –  Qiaochu Yuan May 23 '11 at 12:17
    
@Qiaochu Yuan: No. I mean zero covariance like you said. –  J. J. May 23 '11 at 12:23

1 Answer 1

up vote 4 down vote accepted

Yes, what the exercise claims is true. See Theorem 21 here, and consider the paragraph above it for a proof.

share|improve this answer
    
For general interest, it is interesting to consider mathnet.kaist.ac.kr/mathnet/kms_tex/112250.pdf –  Shai Covo May 23 '11 at 13:19

Your Answer

 
discard

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.