Editing Compound Poisson process (section)

==Exponentiation of measures==
Let ''N'', ''Y'', and ''D'' be as above.  Let ''μ'' be the probability measure according to which ''D'' is distributed, i.e.

:<math>\mu(A) = \Pr(D \in A).\,</math>

Let ''δ''<sub>0</sub> be the trivial probability distribution putting all of the mass at zero.  Then the [[probability distribution]] of ''Y''(''t'') is the measure

:<math>\exp(\lambda t(\mu - \delta_0))\,</math>

where the exponential exp(''ν'') of a finite measure ''ν'' on [[Borel set|Borel subsets]] of the [[real number|real line]] is defined by

:<math>\exp(\nu) = \sum_{n=0}^\infty {\nu^{*n} \over n!}</math>

and

:<math> \nu^{*n} = \underbrace{\nu * \cdots *\nu}_{n \text{ factors}}</math>

is a [[convolution]] of measures, and the series converges [[convergence of random variables|weakly]].