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  "body": "\n## Definition\n\nLet $d \\ge 1$ and let $f \\in L^1(\\R^d)$.\nFor each integer $k \\ge 1$, the *$k$-th convolution power* $f^{*k} \\in L^1(\\R^d)$ is defined by recursion:\n$$f^{*1} = f, \\qquad f^{*(k+1)} = f^{*k} * f,$$\nwhere $*$ is the [[convolution|2Z29JN3VES]].\n\n## Remarks\n\nBy [[associativity of convolution|HZ5J8GGS16]], $f^{*k}$ is the convolution of $k$ copies of $f$ in any bracketing, and $f^{*j} * f^{*k} = f^{*(j+k)}$.\nEach $f^{*k}$ lies in [[$L^1(\\R^d)$|SASP14MA1Z]] with $\\norm{f^{*k}}_1 \\le \\norm{f}_1^k$.\nFor finite Borel measures $\\mu$ the same recursion applies, and one sets $\\mu^{*0} = \\delta_0$, the [[Dirac measure|0ACRGYNHSR]] at $0$, which is the unit for convolution.\n\nExample: for $\\lambda \u003e 0$ let $f(x) = \\lambda e^{-\\lambda x}$ for $x \\ge 0$ and $f(x) = 0$ for $x \u003c 0$, the exponential density.\nThen $f^{*k}(x) = \\lambda^k x^{k-1} e^{-\\lambda x} / (k-1)!$ for $x \\ge 0$.\nThe induction step is\n$$\\int_0^x \\frac{\\lambda^k y^{k-1} e^{-\\lambda y}}{(k-1)!} \\, \\lambda e^{-\\lambda (x - y)} \\, dy = \\frac{\\lambda^{k+1} e^{-\\lambda x}}{(k-1)!} \\int_0^x y^{k-1} \\, dy = \\frac{\\lambda^{k+1} x^k e^{-\\lambda x}}{k!}.$$\nThis is the Erlang density of the sum of $k$ independent exponential waiting times.\n\nNon-example: the constant $f = 1$ on $\\R$ is not in $L^1(\\R)$, and indeed $f * f$ is defined nowhere, since $\\int_\\R 1 \\, dy = \\infty$.\n",
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