Process of repeated convolution, Engineering Mathematics

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If x and y are two independent random variables then their joint density function is given by

The density function f_z of the sum of these two variables is given by the convolution

The proof of this relation may be found in any good introductory text on probability. According to the preceding exercise, if x and y are each U[0,1] then their joint density is triangular, i.e., t = rlr, using symbol t for triangle. What has the central limit theorem got to do with this process of repeated convolution?

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