A note on existence of free Stein kernels

Cébron, Guillaume; Fathi, Max; Mai, Tobias

doi:10.1090/proc/14806

A note on existence of free Stein kernels
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by Guillaume Cébron, Max Fathi and Tobias Mai

Proc. Amer. Math. Soc. 148 (2020), 1583-1594

DOI: https://doi.org/10.1090/proc/14806

Published electronically: November 19, 2019

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Abstract:

Stein kernels are a way of comparing probability distributions, defined via integration by parts formulas. We provide two constructions of Stein kernels in free probability. One is given by an explicit formula, and the other via free Poincaré inequalities. In particular, we show that unlike in the classical setting, free Stein kernels always exist. As corollaries, we derive new bounds on the rate of convergence in the free CLT, and a strengthening of a characterization of the semicircular law due to Biane.

References

M. Banna and T. Mai, Berry-Esseen bounds for the multivariate $\mathcal {B}$-free CLT and operator-valued matrices, in preparation.
Philippe Biane, Logarithmic Sobolev inequalities, matrix models and free entropy, Acta Math. Sin. (Engl. Ser.) 19 (2003), no. 3, 497–506. International Workshop on Operator Algebra and Operator Theory (Linfen, 2001). MR 2014030, DOI 10.1007/s10114-003-0271-5
P. Biane and D. Voiculescu, A free probability analogue of the Wasserstein metric on the trace-state space, Geom. Funct. Anal. 11 (2001), no. 6, 1125–1138. MR 1878316, DOI 10.1007/s00039-001-8226-4
T. Bonis, Rates in the Central Limit Theorem and diffusion approximation via Stein’s Method, Arxiv preprint (2018).
G. Cébron, A quantitative fourth moment theorem in free probability theory, Arxiv preprint (2018).
Sourav Chatterjee, Fluctuations of eigenvalues and second order Poincaré inequalities, Probab. Theory Related Fields 143 (2009), no. 1-2, 1–40. MR 2449121, DOI 10.1007/s00440-007-0118-6
G. P. Chistyakov and F. Götze, Asymptotic expansions in the CLT in free probability, Probab. Theory Related Fields 157 (2013), no. 1-2, 107–156. MR 3101842, DOI 10.1007/s00440-012-0451-2
Thomas A. Courtade, Max Fathi, and Ashwin Pananjady, Existence of Stein kernels under a spectral gap, and discrepancy bounds, Ann. Inst. Henri Poincaré Probab. Stat. 55 (2019), no. 2, 777–790. MR 3949953, DOI 10.1214/18-aihp898
Yoann Dabrowski, A note about proving non-$\Gamma$ under a finite non-microstates free Fisher information assumption, J. Funct. Anal. 258 (2010), no. 11, 3662–3674. MR 2606868, DOI 10.1016/j.jfa.2010.02.010
Max Fathi and Brent Nelson, Free Stein kernels and an improvement of the free logarithmic Sobolev inequality, Adv. Math. 317 (2017), 193–223. MR 3682667, DOI 10.1016/j.aim.2017.06.035
Todd Kemp, Ivan Nourdin, Giovanni Peccati, and Roland Speicher, Wigner chaos and the fourth moment, Ann. Probab. 40 (2012), no. 4, 1577–1635. MR 2978133, DOI 10.1214/11-AOP657
Tobias Mai and Roland Speicher, Operator-valued and multivariate free Berry-Esseen theorems, Limit theorems in probability, statistics and number theory, Springer Proc. Math. Stat., vol. 42, Springer, Heidelberg, 2013, pp. 113–140. MR 3079141, DOI 10.1007/978-3-642-36068-8_{7}
James A. Mingo and Roland Speicher, Free probability and random matrices, Fields Institute Monographs, vol. 35, Springer, New York; Fields Institute for Research in Mathematical Sciences, Toronto, ON, 2017. MR 3585560, DOI 10.1007/978-1-4939-6942-5
Ivan Nourdin and Giovanni Peccati, Normal approximations with Malliavin calculus, Cambridge Tracts in Mathematics, vol. 192, Cambridge University Press, Cambridge, 2012. From Stein’s method to universality. MR 2962301, DOI 10.1017/CBO9781139084659
Nathan Ross, Fundamentals of Stein’s method, Probab. Surv. 8 (2011), 210–293. MR 2861132, DOI 10.1214/11-PS182
Charles Stein, A bound for the error in the normal approximation to the distribution of a sum of dependent random variables, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971) Univ. California Press, Berkeley, Calif., 1972, pp. 583–602. MR 0402873
S. A. Utev, Probabilistic problems connected with an integro-differential inequality, Sibirsk. Mat. Zh. 30 (1989), no. 3, 182–186, 220 (Russian); English transl., Siberian Math. J. 30 (1989), no. 3, 490–493 (1990). MR 1010851, DOI 10.1007/BF00971508
Dan Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory. V. Noncommutative Hilbert transforms, Invent. Math. 132 (1998), no. 1, 189–227. MR 1618636, DOI 10.1007/s002220050222
Alex Zhai, A high-dimensional CLT in $\mathcal W_2$ distance with near optimal convergence rate, Probab. Theory Related Fields 170 (2018), no. 3-4, 821–845. MR 3773801, DOI 10.1007/s00440-017-0771-3