Forward Jump Random Walk on a Cycle Graph and Its Hitting Time

This paper presents an investigation into a random walk on a cycle graph with restricted forward movement at most 𝑚 steps, known as the forward jump random walk. The study derives exact formulas for the probability mass function of the arriving state, the hitting time, and its expected value and variance, where those solutions can be expressed in terms of trigonometric sums. These formulas are obtained using a combinatorial method as an alternative to the eigenvector-based approach commonly used.