Slepian's lemma

For centered multivariate normal distributions represented by vectors $U,V$ with equal corresponding variances and $\mathbb{E}[U_i{U}_j]\ge \mathbb{E}[V_i{V}_j]$ for all distinct indices, $\mathbb{P}(U_i\le u_i\text{ for all }i)\ge \mathbb{P}(V_i\le u_i\text{ for all }i)$ for every threshold vector. In particular the supremum of $U$ is stochastically smaller than the supremum of $V$. Equal corresponding variances are essential to this formulation. The related Sudakov-Fernique inequality compares expected values of maxima using variances of increments and does not require equal pointwise variances.