This definition makes sense if x is an integrable function (in L1), a rapidly decreasing distribution (in particular, a compactly supported distribution) or is a finite Borel measure.
The central limit theorem states that if x is in L1 and L2 with mean zero and variance σ2, then where Φ is the cumulative standard normal distribution on the real line.
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 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).
is an analytic function, then one would like to be able to define If x ∈ L1(Rd) or more generally is a finite Borel measure on Rd, 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 convolutional 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).
Specifically, this holds if x is a compactly supported distribution or lies in the Sobolev space W1,1 to ensure that the derivative is sufficiently regular for the convolution to be well-defined.
In the configuration random graph, the size distribution of connected components can be expressed via the convolution power of the excess degree distribution (Kryven (2017)): Here,
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).
In the formal setting, familiar identities such as continue to hold.
Moreover, by the permanence of functional relations, they hold at the level of functions, provided all expressions are well-defined in an open set by convergent series.