Distances on probability measures and random variables

Abstract

In this paper we lift fundamental topological structures on probability measures and random variables, in particular the weak topology, convergence in law and finite-dimensional convergence to an isometric level. This allows for an isometric quantitative study of important concepts such as relative compactness, tightness, stochastic equicontinuity, Prohorov's theorem and σ -smoothness. In doing so we obtain numerical results which allow for the development of an intrinsic approximation theory and from which moreover all classical topological results follow as easy corollaries.

Fixed points of contractive maps on dcpo's

“…From now on, we will simplify the terminology by omitting the words 'extended' and 'pseudo', so in this respect our terminology differs from common usage, but it does conform with the terminology used in Berckmoes et al (2011) and Colebunders et al (2011), and agrees with the practice in more categorically oriented papers on the subject such as Gutierres and Hofmann (2012). An extended pre-quasi-pseudo metric on a set X is usually a function q : X × X → [0, ∞] that vanishes on the diagonal: if q also satisfies the triangular inequality, it is called an extended quasi-pseudo metric, and if q satisfies both the triangular inequality and symmetry, it is called an extended pseudo metric.…”

“…The approach structure on a continuous dcpo (X, 6) is supposed to induce σ(X). Useful applications of approach theory often rely on this fact -see, for instance, Berckmoes et al (2011) for applications to probability theory. Moreover, as will become clear in the course of this paper, an approach structure deals with convergence of a net or a filter by estimating 'how far' a point is from being a limit point.…”

An application of approach theory to the relative Hausdorff measure of non-compactness for the Wasserstein metric

“…which, by (5) and (13), yields The relative (Hausdorff) measure of non-compactness of a set of probability measures was studied for the weak approach structure in [BLV11], for the continuity approach structure in [B16], and for the parametrized Prokhorov metric in [B16']. …”

“…The structural flexibility of approach theory entails the existence of canonical approach spaces in branches of mathematical analysis such as functional analysis ( [LS00], [SV03], [LV04], [SV04], [SV06], [SV07]), hyperspace theory ( [LS96], [LS98],[LS00']), domain theory ( [CDL11], [CDL14], [CDS14]), and probability theory and statistics ( [BLV11], [BLV13], [BLV16]). A careful study of these approach spaces has resulted in new insights and applications in these branches.…”

Intrinsic irreversibility of Markovian chains

“…. ,N, can be naturally defined in different ways [6][7][8][9][10], with relevant examples being Jeffreys distance [7] and, of course, the Euclidean dis-…”