Compound Poisson process

A compound Poisson process is a continuous-time (random) 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 ${\displaystyle \lambda >0}$ and jump size distribution G, is a process ${\displaystyle \{\,Y(t):t\geq 0\,\}}$ given by

where, ${\displaystyle \{\,N(t):t\geq 0\,\}}$ is 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 ${\displaystyle \{\,N(t):t\geq 0\,\}.\,}$

...
Wikipedia