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 Approximation of stable law in Wasserstein-1 distance by Stein’s method Xu, Lihu1,2 2019-02 Source Publication ANNALS OF APPLIED PROBABILITY ISSN 1050-5164 Volume 29Issue:1Pages:458-504 Abstract Let n is an element of N, let zeta(n,1), . . . , zeta(n,n) be a sequence of independent random variables with E zeta(n,i) = 0 and E vertical bar zeta(n,i)vertical bar < infinity for each i, and let mu be an alpha-stable distribution having characteristic function e(-vertical bar lambda vertical bar alpha) with alpha is an element of (1, 2). Denote S-n = zeta(n,1) + . . . + zeta(n,n) and its distribution by L (S-n), we bound the Wasserstein-1 distance of L(S-n) and mu essentially by an L-1 discrepancy between two kernels. More precisely we prove the following inequality: dW(L(Sn),μ)≤C[∑i=1n∫N−N∣∣∣Kα(t,N)n−Ki(t,N)α∣∣∣dt+RN,n],where dWdW is the Wasserstein-1 distance of probability measures, Kα(t,N)Kα(t,N) is the kernel of a decomposition of the fractional Laplacian Δα2Δα2, Ki(t,N)Ki(t,N) is a KK function (Normal Approximation by Stein’s Method (2011) Springer) with a truncation and RN,nRN,n is a small remainder. The integral term ∑i=1n∫N−N∣∣∣Kα(t,N)n−Ki(t,N)α∣∣∣dt can be interpreted as an L1L1 discrepancy.As an application, we prove a general theorem of stable law convergence rate when ζn,iζn,i are i.i.d. and the distribution falls in the normal domain of attraction of μμ. To test our results, we compare our convergence rates with those known in the literature for four given examples, among which the distribution in the fourth example is not in the normal domain of attraction of μμ. Keyword Stable Approximation Wasserstein-1 Distance (W-1 Distance) Stein's Method L-1 Discrepancy Normal Domain Of Attraction Of Stable Law Alpha-stable Processes DOI http://doi.org/10.1214/18-AAP1424 URL View the original Indexed By SCI Language 英语 WOS Research Area Mathematics WOS Subject Statistics & Probability WOS ID WOS:000452168100011 Publisher INST MATHEMATICAL STATISTICS Fulltext Access Citation statistics Cited Times [WOS]:6   [WOS Record]     [Related Records in WOS] Document Type Journal article Collection DEPARTMENT OF MATHEMATICS Affiliation 1.Univ Macau, Dept Math, Fac Sci & Technol, Av Padre Tomas Pereira, Taipa, Macao, Peoples R China;2.UM Zhuhai Res Inst, Zhuhai, Peoples R China First Author Affilication University of Macau Recommended CitationGB/T 7714 Xu, Lihu. Approximation of stable law in Wasserstein-1 distance by Stein’s method[J]. ANNALS OF APPLIED PROBABILITY,2019,29(1):458-504. APA Xu, Lihu.(2019).Approximation of stable law in Wasserstein-1 distance by Stein’s method.ANNALS OF APPLIED PROBABILITY,29(1),458-504. MLA Xu, Lihu."Approximation of stable law in Wasserstein-1 distance by Stein’s method".ANNALS OF APPLIED PROBABILITY 29.1(2019):458-504.
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