An Erdos-R,v,sz Type Law of the Iterated Logarithm for Order Statistics of a Stationary Gaussian Process

Let be a stationary Gaussian process with almost surely (a.s.) continuous sample paths, and correlation function satisfying (i) as for some and ; (ii) for each and (iii) as for some . For any , consider n mutually independent copies of X and denote by the rth smallest order statistics process, . We provide a tractable criterion for assessing whether, for any positive, non-decreasing function equals 0 or 1. Using this criterion we find, for a family of functions such that , that . Consequently, with , for we have and a.s. Complementarily, we prove an Erdos-R,v,sz type law of the iterated logarithm lower bound on , namely, that a.s. for and a.s. for , where h p( t) = ( 1/ z p( t)) p log log t..