Compound Poisson process

A compound Poisson process is a continuous-time <a href="/facts/Stochastic_process/Ng1n1dnB">stochastic process</a> with jumps. The jumps arrive randomly according to a <a href="/facts/Poisson_process/8uphjAUc">Poisson process</a> 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 
 
 
 
 λ
 >
 0
 
 
 {\displaystyle \lambda >0}
 
 and jump size distribution G, is a process 
 
 
 
 {
 
 Y
 (
 t
 )
 :
 t
 ≥
 0
 
 }
 
 
 {\displaystyle \{\,Y(t):t\geq 0\,\}}
 
 given by

Y
        (
        t
        )
        =
        
          ∑
          
            i
            =
            1
          
          
            N
            (
            t
            )
          
        
        
          D
          
            i
          
        
      
    
    {\displaystyle Y(t)=\sum _{i=1}^{N(t)}D_{i}}

where, 
 
 
 
 {
 
 N
 (
 t
 )
 :
 t
 ≥
 0
 
 }
 
 
 {\displaystyle \{\,N(t):t\geq 0\,\}}
 
 is the counting variable of a <a href="/facts/Poisson_process/8uphjAUc">Poisson process</a> 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
 )
 :
 t
 ≥
 0
 
 }
 .
 
 
 
 {\displaystyle \{\,N(t):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.

Compound Poisson process open-in-new

Compound Poisson process