Randomized Dimension Reduction for Markov Chains Simulation

Nabil Kahale

Published 2017-08-24Version 1

We present an efficient method to estimate a function of the state of a Markov chain at time-step d. Our method is based on a new unbiased algorithm that estimates the expected value of f(U) via Monte Carlo simulation, where U is a vector of d independent random variables, and f is a function that depends little on its last arguments. The algorithm samples an initial segment of components of U according to a certain distribution. For several Markov chains, our method improves upon standard Monte Carlo by a factor asymptotically proportional to d. We establish a relationship between the performance of our algorithm and the truncation dimension of f. The algorithm uses a new geometric method, of independent interest, that computes the optimal distribution in O(d) steps. Numerical experiments support our findings.