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

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