Abstract
This note presents families of inequalities for the Gaussian measure of convex sets which extend the recently proven Gaussian correlation inequality in various directions.
Highlights
Introduction and statement of resultsLet γ be the standard Gaussian on Rn, defined by γ(K) = 1 (2π)n/2 e− 1 2 x dxK for Lebesgue measurable K ⊆ Rn
We present the proof
In Appendix A we provide an interesting reformulation of this monotonicity property in terms of the function sinc x
Summary
For 0 ≤ t ≤ 1 let γt denote the distribution on Rn × Rn of the jointly normal vector (X, Y ) where the distribution of both X and Y is the standard Gaussian measure γ and the covariance matrix is E(XY ) = tI where I is the n × n identity matrix. This family of measures (γt)0≤t≤1 interpolates between γ0(K × L) = γ(K)γ(L) and γ1(K × L) = γ(K ∩ L) and is given explicitly, for t < 1, by the formula γt(H) =. Γ(A)γ(B) ≤ (1 − S)−n/2γt 1 − S(A × B)
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