The discrete renewal theorem with bounded interevent times
Rohan Shenoy
TL;DR
The paper proves the discrete renewal theorem for a 1D process with bounded interarrival times using only elementary real-analysis methods, avoiding Markov-chain theory, complex analysis, or generating functions. It introduces a board-game inspired model, defines visit probabilities $u_n$ and a jump-length distribution with finite support, and derives the renewal equation $u_n = f_n u_0 + f_{n-1}u_1 + \cdots + f_1 u_{n-1}$ along with a tail-based identity. The author then proves that $u_n$ converges to $L = \frac{1}{\operatorname{E}\langle X\rangle}$ by an elementary monotone-convergence argument, and shows that this limit arises from the renewal equation via the tail-sum formula $\operatorname{E}\langle X\rangle = \sum_{k\ge0} P(X>k)$. In the background, the work notes the more general Erdős–Feller–Pollard theorem proved with generating functions and situates the result within general renewal theory, including extensions to Laplace and Fourier transforms for broader settings.
Abstract
The purpose of this note is to prove the celebrated Discrete Renewal Theorem in a common special case. We use only very elementary methods from real analysis, rather than markov chain theory, complex analysis, or generating functions. Provided is an introduction to a 1d discrete renewal process via a board game example, our proof the discrete renewal theorem, as well as background and history of the Erdǒs-Feller-Pollard Theorem.