Convolution power

If $x$ is a function on Euclidean space and $n$ is a natural number, then one can define the convolution power as follows:

x^{*n}\triangleq \underbrace {x*x*x*\cdots *x*x} _{n}

where * denotes the convolution operation. This definition makes sense if x is an integrable function (in L¹), a compactly supported distribution, or is a finite Borel measure.

If μ is a probability measure, then μ is infinitely divisible provided there exists, for each positive integer n, a probability measure μ_1/n such that

\mu _{1/n}^{*n}=\mu .

That is, a measure is infinitely divisible if it is possible to define all nth roots. Not every probability measure is infinitely divisible, and a characterization of infinitely divisible measures is of central importance in the abstract theory of stochastic processes. Intuitively, a measure should be infinitely divisible provided it has a well-defined "convolution logarithm." The natural candidate for measures having such a logarithm are those of (generalized) Poisson type, given in the form

\pi _{\alpha ,\mu }=e^{-\alpha }\sum _{n=0}^{\infty }{\frac {\alpha ^{n}}{n!}}\mu ^{*n}.

In fact, the Lévy–Khinchine theorem states that a necessary and sufficient condition for a measure to be infinitely divisible is that it must lie in the closure, with respect to the vague topology, of the class of Poisson measures (Stroock 1993, §3.2).

Derivatives

If x is itself suitably differentiable, then the properties of convolution, one has

{\mathcal {D}}{\big \{}x^{*n}{\big \}}=({\mathcal {D}}x)*x^{*(n-1)}=x*{\mathcal {D}}{\big \{}x^{*(n-1)}{\big \}}

where ${\mathcal {D}}$ denotes the derivative operator. Specifically, this holds if x is a compactly supported distribution or lies in the Sobolev space W^1,1 to ensure that the derivative is sufficiently regular for the convolution to be well-defined.

References

Stroock, Daniel W. (1993), Probability theory, an analytic view, Cambridge University Press, ISBN 978-0-521-43123-1, MR1267569

Derivatives

See also

References