A compound Poisson process is a continuous-time stochastic process with jumps. The jumps arrive randomly according to a Poisson process and the size of the jumps is also random, with a specified probability distribution. To be precise, a compound Poisson process, parameterised by a rate ${\displaystyle \lambda >0}$ and jump size distribution G, is a process ${\displaystyle \{\,Y(t):t\geq 0\,\}}$ given by

where, ${\displaystyle \{\,N(t):t\geq 0\,\}}$ is the counting variable of a Poisson process with rate ${\displaystyle \lambda }$, and ${\displaystyle \{\,D_{i}:i\geq 1\,\}}$ are independent and identically distributed random variables, with distribution function G, which are also independent of ${\displaystyle \{\,N(t):t\geq 0\,\}.\,}$

When ${\displaystyle D_{i}}$ are non-negative integer-valued random variables, then this compound Poisson process is known as a stuttering Poisson process. ^{[citation needed]}

The expected value of a compound Poisson process can be calculated using a result known as Wald's equation as:

Making similar use of the law of total variance, the variance can be calculated as:

Lastly, using the law of total probability, the moment generating function can be given as follows:

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

Let δ₀ be the trivial probability distribution putting all of the mass at zero. Then the probability distribution of Y(t) is the measure