The supremum of Chi-Square processes

We describe a lower bound for the critical value of the supremum of a ChiSquare process. This bound can be approximated using a MCQMC simulation. We compare numerically this bound with the upper bound given by Davies, only suitable for a regular Chi-Square process. In a second part, we focus a non regular Chi-Square process : the Ornstein-Uhlenbeck Chi-Square process. Recently, Rabier et al. (2009) have shown that this process has an application in genetics : it is the limiting process of the likelihood ratio test process related to the test of a gene on an interval representing a chromosome. Using results from Delong (1981), we propose a theoretical formula for the supremum of such a process and we compare it in particular with our simulated lower bound.