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Approximation of stable law in Wasserstein-1 distance by Stein’s method
Xu, Lihu1,2

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.

KeywordStable Approximation Wasserstein-1 Distance (W-1 Distance) Stein's Method L-1 Discrepancy Normal Domain Of Attraction Of Stable Law Alpha-stable Processes
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Indexed BySCI
WOS Research AreaMathematics
WOS SubjectStatistics & Probability
WOS IDWOS:000452168100011
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Cited Times [WOS]:6   [WOS Record]     [Related Records in WOS]
Document TypeJournal article
Affiliation1.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 AffilicationUniversity of Macau
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GB/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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