Proximity of probability distributions in terms of Fourier-Stieltjes transforms. (English) Zbl 1387.60019

Bobkov, S. G.

Russ. Math. Surv. 71, No. 6, 1021-1079 (2016); translation from Usp. Mat. Nauk 71, No. 6, 37-98 (2016).

This manuscript represents a survey of some results on smoothing inequalities for some probability metrics. Some distances between distribution functions are considered such are Kolmogorov distance, Lévy distance and distance in variation and relations between them are discussed. Some of other considered distances are distances in some metrics, for example \(L^p\)-metric including special case the \(L^1\)-metric, ideal Zolotarev metric and transport metric. At the end of the paper some estimates for the considered distances are given.

Reviewer: Miroslav M. Ristić (Niš)

MSC:

60E05	Probability distributions: general theory
60E10	Characteristic functions; other transforms
60B10	Convergence of probability measures
60F05	Central limit and other weak theorems

Keywords:

probability metrics; smoothing inequalities; Kolmogorov distance; metrics; estimates