Some new normal comparison inequalities related to Gordon’s inequality

Let {ξi,j}{ξi,j} and {ηi,j}{ηi,j}(1≤i≤n,1≤j≤m)(1≤i≤n,1≤j≤m) be standard Gaussian random variables. Gordon’s inequality says that if E(ξi,jξi,k)≥E(ηi,jηi,k)E(ξi,jξi,k)≥E(ηi,jηi,k) for 1≤i≤n,1≤j,k≤m1≤i≤n,1≤j,k≤m, and E(ξi,jξl,k)≤E(ηi,jηl,k)E(ξi,jξl,k)≤E(ηi,jηl,k) for 1≤i≠l≤n,1≤j,k≤m1≤i≠l≤n,1≤j,k≤m, the lower bound P(∪i=1n∩j=1m{ξi,j≤λi,j})/P(∪i=1n∩j=1m{ηi,j≤λi,j}) is at least 1. In this paper, two refinements of upper bound for Gordon’s inequality are given.