APPROXIMATION OF STABLE LAW IN WASSERSTEIN-1 DISTANCE BY STEIN'S METHOD

• 成果类型: Article
• 署名作者: Xu, Lihu
• 署名单位: University of Macau
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/18-AAP1424
• 发表日期: 2019
• 页码: 458-504
• 关键词: exchangeable pairs; invariant-measures; random-variables; CONVERGENCE; rates

摘要

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: d(W)(L(S-n), mu) <= C[Sigma(n )(i=1)integral(N )(-N)vertical bar kappa(alpha)(t, N)/n - K-i(t, N)/alpha vertical bar dt = R-N,R-n], where d(W) is the Wasserstein-1 distance of probability measures, kappa(alpha)(t, N) is the kernel of a decomposition of the fractional Laplacian Delta(alpha/2), K-i(t, N) is a K function (Normal Approximation by Stein's Method (2011) Springer) with a truncation and R-N,R-n is a small remainder. The integral term Sigma(n )(i=1)integral(N )(-N)vertical bar kappa(alpha)(t, N)/n - K-i(t, N)/alpha vertical bar dt can be interpreted as an L-1 discrepancy. As an application, we prove a general theorem of stable law convergence rate when zeta(n)(,i) are i.i.d. and the distribution falls in the normal domain of attraction of mu. 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 mu.