Compositions, inverses and thinnings of random measures

Abstract

Compositions and inverses of measures on the real line are defined as measures whose cumulative distribution functions (c.d.f.'s) are compositions and inverses, respectively, of the c.d.f.'s of the measures involved. We study the continuity of the composition and inverse operators on measures. We then show how a large class of thinnings of point processes and random measures can be characterized by compositions of random measures. We present several convergence theorems for such compositions. These contain, as special cases, the classical thinning theorem of Renyi and many of its contemporary extensions.