Majorants and Minorants for Elliptic Measures on Rk

Let μ(· ; Σ, G1) and μ(· ; Ω, G2) be elliptically contoured measures on Rk centered at 0, having scale parameters (Σ, Ω) and radial cdf′s (G1, G2). Elliptical measures vm(·) and vM(·), depending on (Σ, Ω, G1, G2), are constructed such that Vm(C) ≤ {μ(C; Σ, G1), μ(C; Ω, G2)} for every symmetric convex set C ⊂ Rk with equality for certain sets. These in turn rely on the construction of spectral lower and upper matrix bounds for (Σ, Ω). Extensions include bounds for certain ensembles and mixtures, including versions having star-shaped contours. The lindings specialize to give envelopes for some nonstandard distributions of quadratic forms, with applications to stochastic characteristics of ballistic systems.