Halfplane Trimming for Bivariate Distributions

Let μ be a probability measure on R2 and let u ∈ (0, 1). A bivariate u-trimmed region D(u), defined as the intersection of all halfplanes whose μ-probability measure is at least equal to u, is studied. It is shown that D(u) is not empty for u sufficiently close to 1 and that D(u) satisfies some natural continuity properties. Limit behavior is also considered, the main result being that the weak convergence of a sequence of probability measures entails the pointwise convergence with respect to Hausdorff distance of the associated trimmed regions; this is then applied to derive asymptotics of the empirical trimmed regions. A brief discussion of the extension of the results to higher dimensions is also given.