# Difference between revisions of "Stochastic Processes"

Stochastic Process as taught in Durett's Book in STAT650.

## Poisson Processes

The following 3 definitions of the Poisson process are equivalent.

• A Poisson process with rate ${\displaystyle \lambda }$ is a counting process where the number of arrivals ${\displaystyle N(s+t)-N(s)}$ in a given time period ${\displaystyle t}$ has distribution ${\displaystyle Poisson(\lambda t)}$
• A Poisson process is a renewal process with rate ${\displaystyle 1/\lambda }$
• A Poisson process is a continuous time markov chain with ${\displaystyle P(N(h)=1)=\lambda h+o(h)}$ and ${\displaystyle P(N(h)\geq 2)=o(h)}$