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Let $\tau_1,\tau_2,\dots$ be a sequence of i.i.d. random variables such that $\tau_i \sim Exp(\lambda)$ on $(\Omega,\mathcal E,\mathbb P)$.

Then define $T_i=\sum_{k=1}^i \tau_k$, so we know how each $T_i$ is distributed: $$T_i\sim \Gamma(i,\lambda ).$$

We have a sequence of increasing time $T_1, T_2,\dots$ and then, fixed $\omega \in \Omega$, for each $t\geq 0$ there exists $k=k(\omega)$ such that $T_k(\omega)\leq t < T_{k+1}(\omega)$.

So let $X_t(\omega):=k(\omega)$ and consider the stochastic process $\{X_t\}_t$ on $(\Omega, \mathcal E, \mathbb P)$. Its trajectories are piecewise constant, with jumps in each $T_i$.

Now, using the fact that $T_i\sim \Gamma(i,\lambda)$, we have that $$\mathbb P(X_t=m)=\frac{(\lambda t)^m}{m!}e^{-\lambda t}.$$

Now I have to prove that the increments of this process are independent. But I don't know how to do it.

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    $\begingroup$ Isn't this just a Poisson process? $\endgroup$ – Math1000 Feb 2 '15 at 12:44
  • $\begingroup$ Yes, it is. But I want to prove it explicity. $\endgroup$ – batman Feb 2 '15 at 12:50
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    $\begingroup$ What about a look at the literature? (E.g. Sato's book on Lévy processes.) $\endgroup$ – saz Feb 2 '15 at 13:33
  • $\begingroup$ See for example: columbia.edu/~ks20/stochastic-I/stochastic-I-PP.pdf $\endgroup$ – Math1000 Feb 2 '15 at 14:06

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