# Linear or quadratic combinations of i.i.d. random variables [closed]

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 BukhApr 30 '14 at 19:16

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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