On probabilistic results for the discrepancy of a hybrid-Monte Carlo sequence

Abstract

In many applications it has been observed that hybrid-Monte Carlo sequences perform better than Monte Carlo and quasi-Monte Carlo sequences, especially in difficult problems. For a mixed s -dimensional sequence m , whose elements are vectors obtained by concatenating d -dimensional vectors from a low-discrepancy sequence q with ( s − d ) -dimensional random vectors, probabilistic upper bounds for its star discrepancy have been provided. In a paper of G. Ökten, B. Tuffin and V. Burago [G. Ökten, B. Tuffin, V. Burago, J. Complexity 22 (2006), 435–458] it was shown that for arbitrary ε > 0 the difference of the star discrepancies of the first N points of m and q is bounded by ε with probability at least 1 − 2 exp ( − ε 2 N / 2 ) for N sufficiently large. The authors did not study how large N actually has to be and if and how this actually depends on the parameters s and ε . In this note we derive a lower bound for N , which significantly depends on s and ε . Furthermore, we provide a probabilistic bound for the difference of the star discrepancies of the first N points of m and q , which holds without any restrictions on N . In this sense it improves on the bound of Ökten, Tuffin and Burago and is more helpful in practice, especially for small sample sizes N . We compare this bound to other known bounds.