Finding the Limit

The probability of obtaining an average value $z$ is the product of the probabilities of obtaining arbitrary individual mean values $x_i$, but with the constraint that the average is $z$. We can express this through the following expression

$$\tilde{p}(z)=\int dx_1p(x_1)\int dx_2p(x_2)\dots\int dx_mp(x_m) \delta(z-\frac{x_1+x_2+\dots+x_m}{m}),$$

where the $\delta$-function enbodies the constraint that the mean is $z$. All measurements that lead to each individual $x_i$ are expected to be independent, which in turn means that we can express $\tilde{p}$ as the product of individual $p(x_i)$. The independence assumption is important in the derivation of the central limit theorem.

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