= Slepian's lemma
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= Slepian comparison lemma
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For centered <multivariate normal distributions> represented by vectors $U,V$ with equal corresponding <variances> and $\mathbb E[U_iU_j]\geq\mathbb E[V_iV_j]$ for all distinct indices, $\mathbb P(U_i\leq u_i\text{ for all }i)\geq\mathbb P(V_i\leq u_i\text{ for all }i)$ 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>.
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