Gnedenko-Kolmogorov Central Limit Theorem Revisited
Let be a probability density of the random variable, i.e.
, and .
Suppose that all variables are independent.
The mean and the variance of a given random variable are, respectively
.
The mean and variance of are therefore and .
The density of the random variable corresponding to the sum is given by the
Central Limit Theorem for distributions of compact support (Gnedenko and Kolmogorov) .
Let be distributions such that .
Let, and .
Without loss of generality assume that and .
The random variable holds, as,
where and
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