A normal comparison inequality and its applications

Abstract

Let ξ = (ξi, 1 ≤ i ≤ n) and η = (ηi, 1 ≤ i ≤ n) be standard normal random variables with covariance matrices R1 = (rij1) and R0 = (rij0), respectively. Slepian's lemma says that if rij1 ≥ rij0 for 1 ≤ i, j ≤ n, the lower bound ℙ(ξi ≤ u for 1 ≤ i ≤ n)/ℙ(ηi ≤ u for 1 ≤ i ≤ n) is at least 1. In this paper an upper bound is given. The usefulness of the upper bound is justified with three concrete applications: (i) the new law of the iterated logarithm of Erdos and Révész, (ii) the probability that a random polynomial does not have a real zero and (iii) the random pursuit problem for fractional Brownian particles. In particular, a conjecture of Kesten (1992) on the random pursuit problem for Brownian particles is confirmed, which leads to estimates of principal eigenvalues.