Polynomial-Time Minimax Prediction for Stationary Renewal Processes
Xinyu Chen, Yiming Ma
Abstract
The minimax next-symbol Kullback-Leibler risk for stationary binary renewal processes is known to have order n^(-1/2). We give a polynomial-time predictor attaining this order for an arbitrary unknown finite-mean interarrival law. The construction uses a classical type-uniform renewal code. Its contribution is an exact dynamic program for censored prefix marginals, avoiding enumeration of gap histograms or binary completions. Combined with an established prediction reduction, it gives risk at most [log M_(n+1)+log(n+2)]/n, where M_N counts renewal types. Shared suffix updates reduce the total computation to O(n^3) arithmetic work and polynomial bit complexity, without tail, mixing, known-mean, or bounded-support assumptions. The statistical rate and its lower bound are prior results; the proposed advance concerns computation.