Convolution power

Summary

The $k$-th convolution power $f^{*k}$ of a function or measure $f$ is the convolution of $k$ copies of $f$. It is the analogue of the ordinary power $x^k$ when the product is convolution. If $f$ is the density of a random variable $X$, then $f^{*k}$ is the density of the sum of $k$ independent copies of $X$. Under the Fourier transform, $f^{*k}$ becomes the ordinary power $\hat f^{\,k}$.