Compound Poisson process

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A compound Poisson process is a continuous-time (random) 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. A compound Poisson process, parameterised by a rate and jump size distribution G, is a process given by

where, is a Poisson process with rate , and are independent and identically distributed random variables, with distribution function G, which are also independent of

When are non-negative integer-valued random variables, then this compound Poisson process is known as a stuttering Poisson process which has the feature that two or more events occur in a very short time .

Properties of the compound Poisson process[edit]

Using conditional expectation, 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:

Exponentiation of measures[edit]

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

where the exponential exp(ν) of a finite measure ν on Borel subsets of the real line is defined by


is a convolution of measures, and the series converges weakly.

See also[edit]