Jump Times of the Homogeneous Poisson Process: Finiteness and Exponential Interarrival Law

Let $Y=(Y_u)_{u\ge0}$ be a homogeneous Poisson process with rate $1$ on a probability space $(\Omega,\mathcal{F},P)$, all of whose paths are counting paths, and let $\tau_k=\tau_k(Y)$ denote the $k$-th jump time for each natural number $k\ge1$.

(a) Almost surely: every $\tau_k$ is finite, $0<\tau_1<\tau_2<\dots$, and for every real number $M$ there is a $k$ with $\tau_k>M$.

(b) On the almost sure event of part (a) define the interarrival times $\xi_1=\tau_1$ and $\xi_k=\tau_k-\tau_{k-1}$ for $k\ge2$ (and $\xi_k=0$ off that event). Then for every natural number $k$ and all nonnegative real numbers $u_1,\dots,u_k$,

where $e$ denotes the real exponential function. In particular the interarrival times are independent and each satisfies $P(\xi_k>u)=e^{-u}$ for all $u\ge0$.