---
tag: 8943THTS6W
kind: definition
name: Convolution power
msc: [44A35, 42A85]
defines: JGR1E7Q9KQ
uses: [2Z29JN3VES, SASP14MA1Z, HZ5J8GGS16, 0ACRGYNHSR]
sources:
  - url: https://en.wikipedia.org/wiki/Convolution_power
---

## Definition

Let $d \ge 1$ and let $f \in L^1(\R^d)$.
For each integer $k \ge 1$, the *$k$-th convolution power* $f^{*k} \in L^1(\R^d)$ is defined by recursion:
$$f^{*1} = f, \qquad f^{*(k+1)} = f^{*k} * f,$$
where $*$ is the [[convolution|2Z29JN3VES]].

## Remarks

By [[associativity of convolution|HZ5J8GGS16]], $f^{*k}$ is the convolution of $k$ copies of $f$ in any bracketing, and $f^{*j} * f^{*k} = f^{*(j+k)}$.
Each $f^{*k}$ lies in [[$L^1(\R^d)$|SASP14MA1Z]] with $\norm{f^{*k}}_1 \le \norm{f}_1^k$.
For 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.

Example: for $\lambda > 0$ let $f(x) = \lambda e^{-\lambda x}$ for $x \ge 0$ and $f(x) = 0$ for $x < 0$, the exponential density.
Then $f^{*k}(x) = \lambda^k x^{k-1} e^{-\lambda x} / (k-1)!$ for $x \ge 0$.
The induction step is
$$\int_0^x \frac{\lambda^k y^{k-1} e^{-\lambda y}}{(k-1)!} \, \lambda e^{-\lambda (x - y)} \, dy = \frac{\lambda^{k+1} e^{-\lambda x}}{(k-1)!} \int_0^x y^{k-1} \, dy = \frac{\lambda^{k+1} x^k e^{-\lambda x}}{k!}.$$
This is the Erlang density of the sum of $k$ independent exponential waiting times.

Non-example: the constant $f = 1$ on $\R$ is not in $L^1(\R)$, and indeed $f * f$ is defined nowhere, since $\int_\R 1 \, dy = \infty$.