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I already posted this question here http://math.stackexchange.com/questions/769920/law-of-large-numbers-for-linear-quadratic-combinations-of-i-i-d-random-variab but I received no answers.

Let $(X_i)_{i\in\mathbb{N}}$ be i.i.d. real random variables with zero mean. By the law of large numbers $$\frac{1}{n}\sum_{i=1}^nX_i \to 0 \quad\text{(almost surely, in probabability...) as }\,n\to\infty \;.$$ Now let $(a_i)_{i\in\mathbb{N}}$ be a deterministic real sequence. Under suitable hypothesis (which ones?) is it still true that $$\frac{1}{n}\sum_{i=1}^n a_i X_i \to 0 \quad\text{(at least in probabability) as }\,n\to\infty \;?$$ Furthermore, if $(X_{i,j})_{i,j\in\mathbb{N}}$ is a double indexed sequence of i.i.d. real random variables with zero mean, are there hypothesis such that $$\frac{1}{n^2}\sum_{i,j=1}^n a_i a_j X_{i,j} \to 0 \quad\text{(at least in probabability) as }\,n\to\infty \;?$$

Edited after Igor Rivin's comment.

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closed as off-topic by Bjørn Kjos-Hanssen, Stefan Kohl, Ryan Budney, Andrey Rekalo, Boris Bukh Apr 30 '14 at 19:16

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Thank you, but is there anything about the second question? I didn't find it – user118866 Apr 28 '14 at 11:02
    
The double indexing does not really appear to be relevant, what is relevant is that you have $n^2$ variables, but you are normalizing by $1/n,$ so the variances of your variables better go to zero. How they should go to zero can be found in G/K, or in Feller, v. 2. – Igor Rivin Apr 28 '14 at 14:47
    
Oh sorry! The normalization is wrong, I should put $1/n^2$ – user118866 Apr 30 '14 at 16:10

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