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I am new here so if I violate any rule, please inform.

Consider the stochastic process given by $\{ N(t) : t \geq 0 \}$ which is time homogeneous poisson process with arrival rate $\lambda$. Let $W_n$ be the waiting time, i.e, $W_n = \inf \{t\geq0:N(t)=n \}$.

We want to show that $P(W_k < \infty) = 1 \ \ \forall k=1,2,\ldots $.

Any help would be appreciated.

Thanks.

  • 0
    What would the probability be at infinity?2012-11-25
  • 1
    The result is quite general: each interarrival time is almost surely finite hence the sum $W_k$ of the $k$ first interarrival times is almost surely finite.2012-11-26

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