文摘
Let γγ denote any centered Gaussian measure on RdRd. It is proved that for any closed convex sets AA and BB in RdRd, and any closed convex cones CC and DD in RdRd, if D⊇C∘D⊇C∘, where C∘C∘ is the polar cone of CC, then γ((A+C)∩(B+D))≤γ(A+C)⋅γ(B+D),γ((A+C)∩(B+D))≤γ(A+C)⋅γ(B+D), and γ((A+C)∩(B−D))≥γ(A+C)⋅γ(B−D).γ((A+C)∩(B−D))≥γ(A+C)⋅γ(B−D). As an application, this new inequality is used to bound the asymptotic posterior distributions of likelihood ratio statistics for convex cones.