On the Lower Tail of Gaussian Seminorms

Let E be an infinite-dimensional vector space carrying a Gaussian measure μ with mean 0 and a measurable norm q. Let F(t) := μ(q t). By a result of Borell, F is logarithmically concave. But we show that $F'$ may have infinitely many local maxima for norms q = ₙ|fₙ|/aₙ where fₙ are independent standard normal variables. We also consider Hilbertian norms q = (Σ bₙf²ₙ)1/2 with bₙ > 0, Σ bₙ < ∞. Then as t ↓ 0 we can have F(t) ↓ 0 as rapidly as desired, or as slowly as any function which is o(tⁿ) for all n. For bₙ = 1/n² and in a few closely related cases, we find the exact asymptotic behavior of F at 0. For more general bₙ we find inequalities bounding F between limits which are not too far apart.