A normal comparison inequality and its applications

Let $\xi=(\xi_i, 1 \leq i \leq n)$ and $\eta= (\eta_i, 1 \leq i \leq n)$ be standard normal random variables with covariance matrices $R^1=(r_{ij}^1)$ and $R^0=(r_{ij}^0)$, respectively. Slepian's lemma says that if $r_{ij}^1 \geq r_{ij}^0$ for $1 \leq i, j \leq n$, the lower bound $\P(\xi_i \leq u \mb{for} 1 \leq i \leq n ) /\P(\eta_i \leq u \mb{for} 1 \leq i \leq 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 Erdős 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.