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Let $\mu_i$ be i.i.d. RVs with mean zero, and let $a_i$ be random weights that are not independent and are not identically distributed, $i=1,...,N$. $\mu_i$ is orthogonal to $a_j\;\forall j$.

Is there a law of large numbers (or a set of minimal conditions) that allows the calculation of the probability limit as $N$ goes to infinity of the expression $$\frac1N\sum_{i=1}^N a_i\mu_i$$ ? Is the orthogonality assumption adequate to allow this?

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