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Consider a nonhomogeneous Poisson process $N(t)$ for $t\ge0$ defined by the instantaneous intensity $\lambda(t)\ge0$ and mean value function $\Lambda(t)=\int_0^t\lambda(s)ds$. We know that $$ \mathbb{P}(N(t)=n)=e^{-\Lambda(t)}\frac{\Lambda(t)^n}{n!},\quad n=0,1,\dots. $$ From this we get the CDF of arrival times $T_n$ as $$ F_{T_n}(t)=\mathbb{P}(T_n\le t)=\mathbb{P}(N(t)\ge n)=1-\frac{\Gamma(n,\Lambda(t))}{\Gamma(n)},\quad t\ge0,\,n=1,2,\dots, $$ where the $\Gamma$ functions in the numerator and denominator are the upper incomplete and complete Gamma functions, respectively. From this we get, correctly, $F_{T_n}(0)=0$ as one "normalization" of the CDF irrespective of the choice of $\lambda(t)$.

Question: now suppose either (i) $\lambda(t)=0$ for $t>T$, in which case $\Lambda(\infty)<\infty$, or (ii) $\lambda(t)$ remains positive for all $t\ge0$ but decays strongly for large $t$ such that $\Lambda(\infty)<\infty$. In these cases $$ F_{T_n}(\infty)=1-\frac{\Gamma(n,\Lambda(\infty))}{\Gamma(n)}<1 $$ and the CDF is not normalized. How can one reconcile this? These choices for $\lambda(t)$ seem perfectly valid to me, still, I find it hard to believe an unnormalized CDF/PDF makes sense.

Andras Vanyolos
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  • You do not have to go that far. When $\lambda\equiv 0$ then $N$ never jumps and $\mathbb P(T_n=\infty)=1$ for all $n,.$ In short and in general: the $T_n$ are always $[0,\infty]$ valued which makes it possible that $\lim\limits_{t\to\infty}F_{T_n}(t)<1,.$ The probability we are missing sits in $\infty,.$ – Kurt G. Sep 20 '23 at 14:13
  • @KurtG. Hmm, "always $[0,\infty]$ valued" - interesting. Is there a fundamental principle that requires the inclusion of $\infty$ in the value set and which leads to normalized CDF/PDF after all (if you include the "missing" point mass at $t=\infty$)? Or is it just a "trick" to attribute the missing mass to infinity? E.g. in your example $N$ has only 1 trajectory, $N\equiv0$ for $t\ge0$ and I would not even try to define $T_n$ as it does not make much sense to me. But if you insist, then pushing the point mass of 1 to infinity does indeed normalize $T_n$ but the logic is very unnatural to me – Andras Vanyolos Sep 20 '23 at 16:25
  • It is common practice in the theory of stochastic processes to have stopping times $[0,\infty]$ valued because the events those times describe (jumps in your case) may never occur. – Kurt G. Sep 20 '23 at 16:53
  • Yes, indeed. I worked with first passage times before and they can be infinite when they do not occur. Arrival times are also first passage times, e.g. $T_n$ is the first passage time for the process $N$ to reach level $n$. I think I am at peace with the point mass at infinity. Thanks very much! – Andras Vanyolos Sep 20 '23 at 19:24

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