We're studying renewal processes in my probability class, and I'm a unsure as to where to start with the following practice problem:
$N(t)$ 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 $F(X)=P[X_1≤x]$ is aperiodic with $E[X_1^{k+1}]<∞$.
I'm supposed to first derive a renewal type equation for $E[X_{N(t)+1}^k]$, where $k=1,2,…$ and then compute the limit as $t→∞$ 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!