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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 $$\frac{1}{n^2}\sum_{i,j=1}^n a_i a_j X_{i,j} \to 0 \quad\text{(at least in probabability) as }\,n\to\infty \;?$$

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  • $\begingroup$ Just a proposal: Wouldn't it me more appropriate to look for a law of large numbers of $\frac{\sum_{i=1}^n a_i X_i}{\sum_{k=1}^n a_k}$? Because then it would look like a weighted mean. Otherwise, if you plug in $a_k = k$ or even $a_k = k^2$ wouldn't it be very bad? Or do you know more about your sequence $(a_i)$? $\endgroup$
    – mathie314
    Mar 11, 2015 at 15:16

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