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Suppose there are $K$ distributions $F_1,..,F_K$ and a random variable $x$ such that $E(x | F_i) = \mu_i$. Suppose also that the distributions appear in proportion $p_1 F_1 + ... + p_n F_K$, so that with probability $p_i$, the distribution faced is $F_i$. Is it then the case that $\frac{1}{n} \sum_{i=1}^n x_i \to \sum_{i=1}^K p_i E(x | F_i)$, i.e. does a `mixing' law of large numbers hold?

I have looked for this question elsewhere on stackexchange and have not seen anyone ask it.

user341502
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1 Answers1

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I believe the answer is yes,

$\frac{1}{n} \sum_{i=1}^n x_i = \sum_k \mathbb{1}\{\text{dist. faced is $F_k$}\}\frac{1}{n} \sum_{i=1}^n x_i \to \sum_k p_k E(x | F_k)$ where the last part follows by law of large numbers for each of the $k$ distributions.

user341502
  • 513