# The pdf of last interval of a Poisson process in $[0,T]$

Assume a Poisson point process with rate $\lambda$ in time $[0,T]$. Supoose $X$ is the random variable representing the time between the last arrival and $T$. What is the probability density function of $X$ as $T\to \infty$?

The pdf is $\frac{d}{dx}P\left(X\leq x\right)$. We can break up $P\left(X\leq x\right)$ by the number of arrivals in time $[0,T]$:

\begin{align} P\left(X\leq x\right)&=\sum_{k=1}^{\infty}Poisson(k, \lambda T)P\left(X\leq x|k \text{ arrivals}\right)\\ &=\sum_{k=1}^{\infty}Poisson(k, \lambda T)P\left(T-T_k\leq x|k \text{ arrivals}\right)\\ &=\sum_{k=1}^{\infty}Poisson(k, \lambda T)P\left(T-T_k\leq x|T_k \leq T <T_{k+1}\right) \end{align}

where $T_k=\sum_{i=1}^{k}A_i$ is the time $k^{th}$ arrival, and $A_i$ is the $i^{th}$ inter-arrival time. Any idea how to continue? Or, a resource that already has the answer?

• If you were to omit \left and \right in something like $\displaystyle (\sum_{i=1}^n \frac{x_i}{y_i})$ then that is what you would see instead of $\displaystyle \left(\sum_{i=1}^n \frac{x_i}{y_i} \right),$ but it seems to me that in things like $\Pr(X\le x)$ including \left and \right instead of just coding it as \Pr(X\le x) just makes the code cluttered. – Michael Hardy Feb 8 '17 at 19:05
• How is $\lambda T$ approaching $\infty$? Do you mean $T$ approaches $\infty$ with $\lambda$ fixed, or $\lambda$ approaches $\infty$ with $T$ fixed, or something else? Those situations are quite different from each other. – Michael Hardy Feb 8 '17 at 19:08
• @MichaelHardy $T \to \infty$. I edited the problem statement. – Sus20200 Feb 8 '17 at 19:14
• It is explained in many places that, defining $X=T$ if there is no arrival at all in $[0,T]$ (or, equivalently, considering that there is always an arrival at time $0$), one gets that $X$ is distributed like $\min(T,A)$, where $A$ is exponentially distributed with parameter $\lambda$. I will let you deduce the limit in distribution of $X$ when $T\to\infty$. – Did Apr 12 '17 at 12:01
• @Did Can you please let me know some examples of resources that explain as you said? – Sus20200 Apr 12 '17 at 21:41