convolution of probability measures

Question

What do we mean by convolution of measures? With example

What is the difference between convolution of measures and convolution of functions?

What is probability measure?

Give an example of convolution of probability measure ?

Answer 1

If $\mu$ and $\nu$ are two measures (over $\mathbb{R}$ for example), the convolution of $\mu$ and $\nu$ is defined, by $$(\mu\ast\nu)(A)=\int_{\mathbb{R}^2}\mathbf{1}_A(x+y)d\mu(x)d\nu(y).$$

for any measurable set $A$. So the convolution of two measures is a measure, not a function.

A probability measure is a measure on $\Omega$, such that $\mu(\Omega)=1$.

For example, we use convolution of measures in probability theory. If $X$ has distribution $P$ and $Y$ has distribution $Q$, $X$ independent from $Y$, then the distribution of $X+Y$ is $P\ast Q$.

A very useful special case is when $P$ and $Q$ are absolutely continuous w.r.t. the Lebesgue measure, i.e. $dP(x)=f(x)dx$ and $dQ(y)=g(y)dy$. In that case $P\ast Q$ has a density which is the convolution of the two densities: $f\ast g$ (this time it's a convolution of functions, which results in a function). So $d(P\ast Q)(z)=(f\ast g)(z)dz$, where $$(f\ast g)(z) = \int_{\mathbb{R}}f(x)g(z-x)dx = \int_{\mathbb{R}}f(z-y)g(y)dy.$$