# Stochastic Processes

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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 $$\displaystyle P(N(h) \geq 2) = o(h)$$