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 | |
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 Citation 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. |
Files in This Item: | ||||||
There are no files associated with this item. |
Items in the repository are protected by copyright, with all rights reserved, unless otherwise indicated.
Edit Comment