4:25 AM
@SiongThyeGoh can you see pg 169 of the book karlin, how did he write the $F_{n}(x)$ like that ?

4:47 AM
they are conditioning on the last event.

My thinking says that this is again that convolution thingy!!!!

$F_{n}$ as the convolution of $F_{n-1}$ and $F_{1}$

$$S_n = S_{n-1}+X_n$$

Yes and as $F_{n}(x) = P(S_{n} \leq x)$
sorry it must be $F_{n}(t) = P(S_{n} \leq t)$

4:57 AM
\begin{align} F_n(x) &=Pr(S_n \le x) \\&= \int_0^\infty Pr(S_{n-1}+X_n \le x| X_n = y)\, dF(y) \\ &= \int_0^\infty Pr(S_{n-1} \le x-y| X_n = y)\, dF(y)\\&=\int_0^\infty Pr(S_{n-1}\le x-y)\, dF(y) \end{align}

5:07 AM
Wow nice I see

5:32 AM
Observe that the excess life at time t exceeds x if and only if there are
n o renewals in the interval (t, t +x] (Figure 3). @SiongThyeGoh ?
pg 173 ?

I dont understand that statement :'(

excess life is time until next event

after an event at time $t$ we have the very next event only at time $t+x$
nd not between $t$ and $t+x$ ?
is this what that means?

x is jsut a number

5:37 AM
not getting sorry

it's like the event that I will live for more than x more years means death comes after x years
excess life at time $t$ exceeds $x$ means ( I am $t$ age years old now, I get to live for another $x$ years) if and only there are no renewal in the interval $(t, t+x]$ (there is no death within $(t, t+x]$ )

@SiongThyeGoh Very nice this is :)!!
gotcha
and perhaps due to homeogeneity, memorylessness of Poisson process the event has same probability in $(0,x]$
?

yup
i call it stationary
maybe some people use other terms

perhaps it is correct to use stationary implying independent of the time but depends on the step size
and no renewals in $(t,t+x]$ implies $N(t+x) - N(t) = 0$