Let mu be a probability measure on R2 and let u is-an-element-of (0, 1
). A bivariate u-trimmed region D(u), defined as the intersection of a
ll halfplanes whose mu-probability measure is at least equal to u, is
studied. It is shown that D(u) is not empty for u sufficiently close t
o 1 and that D(u) satisfies some natural continuity properties. Limit
behavior is also considered, the main result being that the weak conve
rgence of a sequence of probability measures entails the pointwise con
vergence with respect to Hausdorff distance of the associated trimmed
regions; this is then applied to derive asymptotics of the empirical t
rimmed regions. A brief discussion of the extension of the results to
higher dimensions is also given. (C) 1994 Academic Press, Inc.