Investigation on the Convergence Properties of a Markov Chain

This study investigates the convergence properties of a recursive sequence defined by \(s_0 = 2\) and \(s_{i+1} = 2R(1, s_i)\) , where R(a, b) is a random integer uniformly distributed between a and b. We aim to determine the expected number of steps for the sequence to exceed a threshold n, denoted E(n). Via forming a system of equations from probabilistic analysis and the use of matrices, the research is able to form equations to predict E(n) for large n. Computational experimenting reveals a linear trend as n tends to infinity, with an asymptotic approximation of \(E(n) \approx 2.51778n\) . We also explored an alternative approach involving binary partitions, a well-known function studied by Euler. Through this research, we gained insight on the hidden patterns lying behind this sequence and its connections to Markov chains, with potential applications in randomized algorithms and recursive systems.