Convolution of Measures
Let G be a topological group. If μ and ν are finite Borel measures on G, then their convolution μ∗ν is defined by
for each measurable subset E of G. The convolution is also a finite measure, whose total variation satisfies
In the case when G is locally compact with (left-)Haar measure λ, and μ and ν are absolutely continuous with respect to a λ, so that each has a density function, then the convolution μ∗ν is also absolutely continuous, and its density function is just the convolution of the two separate density functions.
If μ and ν are probability measures on the topological group (R,+), then the convolution μ∗ν is the probability distribution of the sum X + Y of two independent random variables X and Y whose respective distributions are μ and ν.
Read more about this topic: Convolution
Famous quotes containing the word measures:
“the dread
That how we live measures our own nature,
And at his age having no more to show
Than one hired box should make him pretty sure
He warranted no better,”
—Philip Larkin (19221985)