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=supn|fn|/an where fn are independent standard normal variables. We also consider Hilbertian norms q=(Σbnf2n)12 with bn>0,Σbn<∞. 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.