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  "body": "\n## Summary\n\nThe $k$-th convolution power $f^{*k}$ of a function or measure $f$ is the [[convolution|2Z29JN3VES]] of $k$ copies of $f$.\nIt is the analogue of the ordinary power $x^k$ when the product is convolution.\nIf $f$ is the density of a random variable $X$, then $f^{*k}$ is the density of the sum of $k$ independent copies of $X$.\nUnder the Fourier transform, $f^{*k}$ becomes the ordinary power $\\hat f^{\\,k}$.\n\n## Explanations\n\n### L1 human\n\nConvolving a function with itself again and again gives its convolution powers.\nFor the uniform density on $[0, 1]$, the second power is a tent on $[0, 2]$, the third is a rounded hump on $[0, 3]$ made of quadratic pieces, and so on.\nThe higher powers look more and more like a bell curve, which is the central limit theorem in action.\n",
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