Renewal equation ($u_n={\sum }_{k=1}^n{f}_k{u}_{n-k},u_0=1$)

A renewal equation decomposes a process at its first renewal. For returns of a Markov chain to a fixed state, $f_k$ is the first-return probability and $u_n$ the probability of being at that state at time $n$. Their probability generating functions satisfy $U=1+FU$.

For positive interarrivals with law $K$, the unique locally bounded solution of $z=g+z\ast K$ is $z=g\ast U={\sum }_{j\ge 0}g\ast K^{\ast j}$, where $U$ is the renewal measure. Iteration proves the identity: the residual is bounded on each compact interval by a constant times the probability that the sum of many positive interarrivals stays in that interval, which tends to zero.

A defective renewal equation is $z(u)=g(u)+{\int }_{[0,u]}z(u-x)Q(dx)$ where the nonnegative kernel measure has total mass less than one. Its renewal representation is a convergent sum of convolutions. Exponential tilting can turn the kernel into a probability measure, enabling the key renewal theorem.