We consider definition of Poisson processes that satisfy Condition 0 and 1 according to Billingsley section 23 . How find out the densities of $A_t , B_t , L_t$ defined as in problems section of Section 23 ?
Let $X_1$, be the waiting time to the first event, let X 2 be the waiting time between the first and second events, and so on. The formal model consists of an infinite sequence $X_1 , X_2 , ... $ of random variables on some probability space, and $S$, $r = X_1 + \cdots +X_n$ represents the time of occurrence of the nth event; it is convenient to write $S_0 = 0$
Define $N_t=\max[n:S_n <t]$
Let $A_t = t - S_{N_t}$ he the time back to the most recent event in the Poisson stream (or to 0), and let $B_t = S_{N_t+1} — t$ be the time forward to the next event. Show that A, and B, are independent, that $B$, is distributed as $X_1$ (exponentially with parameter $a$), and that $A$, is distributed as $\min{(X_1 , t)}$
Let $L_t = A_t + B_t = S_{N_t+1} - S_{N_t}$ be the length of the interarrival interval covering $t$.
For each $ \omega$ , $\sup_n S_n(\omega) = \infty$ and $\displaystyle\sum_n X_n = \infty$ . The $X_n$ are independent, and each is exponentially distributed with parameter $a$.
For each $\omega$ , $N_t(\omega)$ is a nonnegative integer for $t \geq 0$, $N_o (\omega) = 0$, and $Lim_t N_t(\omega) = \infty$; further, for each $\omega$, $N_t(\omega)$ as a function of $t$ is nondecreasing and right-continuous, and at the points of discontinuity are $N_t(\omega)$ - $\sup_{s < t} N_s (\omega) $ is exactly 1