1
$\begingroup$

When proving the renewal equation for a renewal process in Wikipedia

The renewal function satisfies $$ m(t) = F_S(t) + \int_0^t m(t-s) f_S(s)\, ds $$ where $F_S$ is the cumulative distribution function of $S_1$ and $f_S$ is the corresponding probability density function.

Proof of the renewal equation

We may iterate the expectation about the first holding time: $$ m(t) = \mathbb{E}[X_t] = \mathbb{E}[\mathbb{E}(X_t \mid S_1)]. \, $$ But by the Markov property $$ \mathbb{E}(X_t \mid S_1=s) = \mathbb{I}_{\{t \geq s\}} \left( 1 + \mathbb{E}[X_{t-s}] \right). \, $$ ...

A renewal process is not Markovian. Why is there "by the Markov property" in the proof? Thanks!

$\endgroup$
1
$\begingroup$

The renewal process $(X_t)_{t\geqslant0}$ is not a Markov process but it satisfies the Markov property at time $S_1$. That is, for every $s$, the process $(X_{t+s})_{t\geqslant0}$ conditioned by the event $[S_1=s]$ is distributed like the process $(1+X_t)_{t\geqslant0}$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.