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This is Exercise 2.1 of Varadhan's Stochastic Processes. Let $\tau_i$ be a sequence of independent identically distributed random variables with a common exponential distribution $e^{-\tau}\textrm{d}\tau$. Define

$N(t) = 0$ if $0\leq t < \tau_1 $

$N(t)= k$ if $\tau_1+\ldots+\tau_k \leq t < \tau_1+\ldots+\tau_{k+1}$.

I'm trying to show that $N(t)$ is a Poisson process and for this, I need to show that the distribution of $N(t+h)-N(t)$ is independent of t,i.e. the same for all $t$. Thanks for any help.

I'll try to give an answer to the above question. $N(t)$ and $N(t+h) - N(t)$ are independent. $P(N(t+h) - N(t)=n) = P(N(t+h) - N(t)=n\ \vert N(t)=k )= $ $ P(\tau_{k+1}+\ldots+\tau_{k+n}\leq h < \tau_{k+1}+\ldots+\tau_{k+n+1})$ is independent of $t$. In fact, it is the Poisson distribution of rate $\lambda h$ as mentioned by Kurt G. in the comments below.

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  • Express the events ${N(t+h)-N(t)=n},,{N(t)=k}$ in terms of $\tau_i,.$ – Kurt G. Nov 21 '23 at 18:47
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    We are not trying to show that the event ${\tau_1+\dots+\tau_n\le h<\tau_1+\dots+\tau_{n+1}}$ is independent of $t,.$ After thinking much about this exercise I believe that you should follow this route. – Kurt G. Nov 22 '23 at 16:34

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