We're studying renewal processes in my probability class, and I'm a little unsure as to where to start with the following practice problem we were given:
$N(t)$ is a renewal process and where $X_{N(t)+1}$ is the length of the renewal interval containing point t. Assume that the underlying interarrival distribution $F(X) = P[X_1 \leq x]$ is aperiodic with $E[X_1^{k+1}] < \infty$.
I'm supposed to first derive a renewal type equation for $E[(X_{N(t)+1})^k]$, where $k = 1, 2, \dots$ and then compute the limit as $t \rightarrow \infty$ of $E[(X_{N(t)+1})^k]$ using the key renewal theorem. Would appreciate if someone could walk me through the solution or the solution framework! Thanks!