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
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.
Let δ0 be the trivial probability distribution putting all of the mass at zero. Then the probability distribution of Y(t) is the measure
Fitting a compound Poisson process
- Poisson process
- Poisson distribution
- Non-homogeneous Poisson process
- Fractional Poisson process
- Campbell's formula for the moment generating function of a compound Poisson process
- Simar, L. (1976). "Maximum Likelihood Estimation of a Compound Poisson Process". The Annals of Statistics 4 (6): 1200. doi:10.1214/aos/1176343651. JSTOR 2958588.
- Böhning, D. (1982). "Convergence of Simar's Algorithm for Finding the Maximum Likelihood Estimate of a Compound Poisson Process". The Annals of Statistics 10 (3): 1006. doi:10.1214/aos/1176345890. JSTOR 2240923.