Stochastic Processes

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

Markov Chains

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

Renewal Processes

Queueing Theory