## The Annals of Applied Probability

### Approximation of stable law in Wasserstein-1 distance by Stein’s method

Lihu Xu

#### Abstract

Let $n\in\mathbb{N}$, let $\zeta_{n,1},\ldots,\zeta_{n,n}$ be a sequence of independent random variables with $\mathbb{E}\zeta_{n,i}=0$ and $\mathbb{E}|\zeta_{n,i}|<\infty$ for each $i$, and let $\mu$ be an $\alpha$-stable distribution having characteristic function $e^{-|\lambda|^{\alpha}}$ with $\alpha\in(1,2)$. Denote $S_{n}=\zeta_{n,1}+\cdots+\zeta_{n,n}$ and its distribution by $\mathcal{L}(S_{n})$, we bound the Wasserstein-1 distance of $\mathcal{L} (S_{n})$ and $\mu$ essentially by an $L^{1}$ discrepancy between two kernels. More precisely, we prove the following inequality: $d_{W}\big(\mathcal{L}(S_{n}),\mu\big)\le C\Bigg[\sum_{i=1}^{n}\int_{-N}^{N}\bigg\vert \frac{\mathcal{K}_{\alpha}(t,N)}{n}-\frac{K_{i}(t,N)}{\alpha}\bigg\vert \,\mathrm{d}t+\mathcal{R}_{N,n}\Bigg],$ where $d_{W}$ is the Wasserstein-1 distance of probability measures, $\mathcal{K}_{\alpha}(t,N)$ is the kernel of a decomposition of the fractional Laplacian $\Delta^{\frac{\alpha}{2}}$, $K_{i}(t,N)$ is a $K$ function (Normal Approximation by Stein’s Method (2011) Springer) with a truncation and $\mathcal{R}_{N,n}$ is a small remainder. The integral term $\sum_{i=1}^{n}\int_{-N}^{N}\bigg\vert \frac{\mathcal{K}_{\alpha}(t,N)}{n}-\frac{K_{i}(t,N)}{\alpha}\bigg\vert \,\mathrm{d}t$ 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 $\mu$.

#### Article information

Source
Ann. Appl. Probab., Volume 29, Number 1 (2019), 458-504.

Dates
Revised: August 2018
First available in Project Euclid: 5 December 2018

https://projecteuclid.org/euclid.aoap/1544000434

Digital Object Identifier
doi:10.1214/18-AAP1424

Mathematical Reviews number (MathSciNet)
MR3910009

Zentralblatt MATH identifier
07039130

#### Citation

Xu, Lihu. Approximation of stable law in Wasserstein-1 distance by Stein’s method. Ann. Appl. Probab. 29 (2019), no. 1, 458--504. doi:10.1214/18-AAP1424. https://projecteuclid.org/euclid.aoap/1544000434

#### References

• [1] Afendras, G., Papadatos, N. and Papathanasiou, V. (2011). An extended Stein-type covariance identity for the Pearson family with applications to lower variance bounds. Bernoulli 17 507–529.
• [2] Albeverio, S., Rüdiger, B. and Wu, J.-L. (2000). Invariant measures and symmetry property of Lévy type operators. Potential Anal. 13 147–168.
• [3] Arras, B. and Houdré, C. On Stein’s method for infinitely divisible laws with finite first moment. Available at arXiv:1712.10051.
• [4] Arras, B., Mijoule, G., Poly, G. and Swan, Y. A new approach to the Stein–Tikhomirov method: With applications to the second Wiener chaos and Dickman convergence. Available at arXiv:1605.06819.
• [5] Barbour, A. D., Čekanavičius, V. and Xia, A. (2007). On Stein’s method and perturbations. ALEA Lat. Am. J. Probab. Math. Stat. 3 31–53.
• [6] Barbour, A. D., Gan, H. L. and Xia, A. (2015). Stein factors for negative binomial approximation in Wasserstein distance. Bernoulli 21 1002–1013.
• [7] Baxter, G. and Shapiro, J. M. (1960). On bounded infinitely divisible random variables. Sankhyā 22 253–260.
• [8] Boonyasombut, V. and Shapiro, J. M. (1970). The accuracy of infinitely divisible approximations to sums of independent variables with application to stable laws. Ann. Math. Stat. 41 237–250.
• [9] Braverman, A. and Dai, J. G. (2017). Stein’s method for steady-state diffusion approximations of $M/Ph/n+M$ systems. Ann. Appl. Probab. 27 550–581.
• [10] Butzer, P. L. and Hahn, L. (1978). General theorems on rates of convergence in distribution of random variables. II. Applications to the stable limit laws and weak law of large numbers. J. Multivariate Anal. 8 202–221.
• [11] Chatterjee, S. (2007). Stein’s method for concentration inequalities. Probab. Theory Related Fields 138 305–321.
• [12] Chatterjee, S. (2009). Fluctuations of eigenvalues and second order Poincaré inequalities. Probab. Theory Related Fields 143 1–40.
• [13] Chatterjee, S. and Shao, Q.-M. (2011). Nonnormal approximation by Stein’s method of exchangeable pairs with application to the Curie–Weiss model. Ann. Appl. Probab. 21 464–483.
• [14] Chen, L. H. Y. (1975). Poisson approximation for dependent trials. Ann. Probab. 3 534–545.
• [15] Chen, L. H. Y., Goldstein, L. and Shao, Q.-M. (2011). Normal Approximation by Stein’s Method. Springer, Heidelberg.
• [16] Chen, Z.-Q. and Kumagai, T. (2003). Heat kernel estimates for stable-like processes on $d$-sets. Stochastic Process. Appl. 108 27–62.
• [17] Chen, Z.-Q. and Wang, J. (2014). Ergodicity for time-changed symmetric stable processes. Stochastic Process. Appl. 124 2799–2823.
• [18] Chen, Z.-Q. and Zhang, X. (2016). Heat kernels and analyticity of non-symmetric jump diffusion semigroups. Probab. Theory Related Fields 165 267–312.
• [19] Christoph, G. and Wolf, W. (1992). Convergence Theorems with a Stable Limit Law. Mathematical Research 70. Akademie-Verlag, Berlin.
• [20] Davydov, Yu. and Nagaev, A. V. (2002). On two approaches to approximation of multidimensional stable laws. J. Multivariate Anal. 82 210–239.
• [21] Döbler, C. (2015). Stein’s method of exchangeable pairs for the beta distribution and generalizations. Electron. J. Probab. 20 no. 109, 34.
• [22] Durrett, R. (2010). Probability: Theory and Examples, 4th ed. Cambridge Series in Statistical and Probabilistic Mathematics 31. Cambridge Univ. Press, Cambridge.
• [23] Eichelsbacher, P. and Löwe, M. (2010). Stein’s method for dependent random variables occurring in statistical mechanics. Electron. J. Probab. 15 962–988.
• [24] Fang, X. (2014). Discretized normal approximation by Stein’s method. Bernoulli 20 1404–1431.
• [25] Gaunt, R. E., Pickett, A. M. and Reinert, G. (2017). Chi-square approximation by Stein’s method with application to Pearson’s statistic. Ann. Appl. Probab. 27 720–756.
• [26] Goldstein, L. and Reinert, G. (1997). Stein’s method and the zero bias transformation with application to simple random sampling. Ann. Appl. Probab. 7 935–952.
• [27] Götze, F. and Tikhomirov, A. N. (2006). Limit theorems for spectra of random matrices with martingale structure. Teor. Veroyatn. Primen. 51 171–192.
• [28] Hall, P. (1981). On the rate of convergence to a stable law. J. Lond. Math. Soc. (2) 23 179–192.
• [29] Hall, P. (1981). Two-sided bounds on the rate of convergence to a stable law. Z. Wahrsch. Verw. Gebiete 57 349–364.
• [30] Häusler, E. and Luschgy, H. (2015). Stable Convergence and Stable Limit Theorems. Probability Theory and Stochastic Modelling 74. Springer, Cham.
• [31] Hsu, E. P. (2005). Characterization of Brownian motion on manifolds through integration by parts. In Stein’s Method and Applications. Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. 5 195–208. Singapore Univ. Press, Singapore.
• [32] Johnson, O. and Samworth, R. (2005). Central limit theorem and convergence to stable laws in Mallows distance. Bernoulli 11 829–845.
• [33] Kolokoltsov, V. (2000). Symmetric stable laws and stable-like jump-diffusions. Proc. Lond. Math. Soc. (3) 80 725–768.
• [34] Kuske, R. and Keller, J. B. (2000/01). Rate of convergence to a stable law. SIAM J. Appl. Math. 61 1308–1323.
• [35] Kusuoka, S. and Tudor, C. A. (2012). Stein’s method for invariant measures of diffusions via Malliavin calculus. Stochastic Process. Appl. 122 1627–1651.
• [36] Kwaśnicki, M. (2017). Ten equivalent definitions of the fractional Laplace operator. Fract. Calc. Appl. Anal. 20 7–51.
• [37] Ledoux, M., Nourdin, I. and Peccati, G. (2015). Stein’s method, logarithmic Sobolev and transport inequalities. Geom. Funct. Anal. 25 256–306.
• [38] Ley, C., Reinert, G. and Swan, Y. (2017). Stein’s method for comparison of univariate distributions. Probab. Surv. 14 1–52.
• [39] Nourdin, I. and Peccati, G. (2009). Stein’s method on Wiener chaos. Probab. Theory Related Fields 145 75–118.
• [40] Nourdin, I. and Peccati, G. (2012). Normal Approximations with Malliavin Calculus: From Stein’s Method to Universality. Cambridge Tracts in Mathematics 192. Cambridge Univ. Press, Cambridge.
• [41] Nourdin, I., Peccati, G. and Swan, Y. (2014). Entropy and the fourth moment phenomenon. J. Funct. Anal. 266 3170–3207.
• [42] Partington, J. R. (2004). Linear Operators and Linear Systems: An Analytical Approach to Control Theory. London Mathematical Society Student Texts 60. Cambridge Univ. Press, Cambridge.
• [43] Peköz, E. A., Röllin, A. and Ross, N. (2013). Degree asymptotics with rates for preferential attachment random graphs. Ann. Appl. Probab. 23 1188–1218.
• [44] Reinert, G. and Röllin, A. (2009). Multivariate normal approximation with Stein’s method of exchangeable pairs under a general linearity condition. Ann. Probab. 37 2150–2173.
• [45] Stein, C. (1972). A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. II: Probability theory 583–602. Univ. Calififornia Press, Berkeley, CA.
• [46] Stein, E. M. and Shakarchi, R. (2003). Fourier Analysis: An Introduction. Princeton Lectures in Analysis 1. Princeton Univ. Press, Princeton, NJ.
• [47] Uchaikin, V. V. and Zolotarev, V. M. (1999). Chance and Stability: Stable Distributions and Their Applications. VSP, Utrecht.
• [48] Yuozulinas, A. and Paulauskas, V. (1998). Some remarks on the rate of convergence to stable laws. Liet. Mat. Rink. 38 439–455.