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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. A compound Poisson process, parameterised by a rate λ > 0 {\displaystyle \lambda >0} and jump size distribution G, is a process { Y : t ≥ 0 } {\displaystyle \{\,Y:t\geq 0\,\}} given by
where, { N : t ≥ 0 } {\displaystyle \{\,N:t\geq 0\,\}} is a counting of a Poisson process with rate λ {\displaystyle \lambda } , and { D i : i ≥ 1 } {\displaystyle \{\,D_{i}:i\geq 1\,\}} are independent and identically distributed random variables, with distribution function G, which are also independent of { N : t ≥ 0 } . {\displaystyle \{\,N:t\geq 0\,\}.\,}
When D i {\displaystyle D_{i}} 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.