 Convolution power

In mathematics, the convolution power is the nfold iteration of the convolution with itself. Thus if x is a function on Euclidean space R^{d} and n is a natural number, then the convolution power is defined by
where * denotes the convolution operation of functions on R^{d} and δ_{0} is the Dirac delta distribution. This definition makes sense if x is an integrable function (in L^{1}), a compactly supported distribution, or is a finite Borel measure.
If x is the distribution function of a random variable on the real line, then the n^{th} convolution power of x gives the distribution function of the sum of n independent random variables with identical distribution x. The central limit theorem states that if x is in L^{1} and L^{2} with mean zero and variance σ^{2}, then
where Φ is the cumulative standard normal distribution on the real line. Equivalently, tends weakly to the standard normal distribution.
In some cases, it is possible to define powers x^{*t} for arbitrary real t > 0. 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
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 welldefined "convolution logarithm." The natural candidate for measures having such a logarithm are those of (generalized) Poisson type, given in the form
In fact, the Lévy–Khinchin 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).
Many applications of the convolution power rely on being able to define the analog of analytic functions as formal power series with powers replaced instead by the convolution power. Thus if is an analytic function, then one would like to be able to define
If x ∈ L^{1}(R^{d}) or more generally is a finite Borel measure on R^{d}, then the latter series converges absolutely in norm provided that the norm of x is less than the radius of convergence of the original series defining F(z). In particular, it is possible for such measures to define the complex exponential
It is not generally possible to extend this definition to arbitrary distributions, although a class of distributions on which this series still converges in an appropriate weak sense is identified by Ben Chrouda, El Oued & Ouerdiane (2002).
As convolution algebras are special cases of Hopf algebras, the convolution power is a special case of the (ordinary) power in a Hopf algebra. In applications to quantum field theory, the convolution exponential, convolution logarithm, and other analytic functions based on the convolution are constructed as formal power series in the elements of the algebra (Brouder, Frabetti & Patras 2008). If, in addition, the algebra is a Banach algebra, then convergence of the series can be determined as above. In the formal setting, familiar identities such as
 x = log ^{*} (exp ^{*} x) = exp ^{*} (log ^{*} x)
continue to hold. Moreover, by the permanence of functional relations, they hold at the level of functions, provided all expressions are welldefined in an open set by convergent series.
Properties
If x is itself suitably differentiable, then the properties of convolution, one has
where 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 welldefined.
See also
References
 Ben Chrouda, Mohamed; El Oued, Mohamed; Ouerdiane, Habib (2002), "Convolution calculus and applications to stochastic differential equations", Soochow Journal of Mathematics 28 (4): 375–388, ISSN 02503255, MR1953702.
 Brouder, Christian; Frabetti, Alessandra; Patras, Frédéric (2008). "Decomposition into oneparticle irreducible Green functions in manybody physics". arXiv:0803.3747..
 Feller, William (1971), An introduction to probability theory and its applications. Vol. II., Second edition, New York: John Wiley & Sons, MR0270403.
 Stroock, Daniel W. (1993), Probability theory, an analytic view, Cambridge University Press, ISBN 9780521431231, MR1267569.
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