Abstract：We review recent advances in Stein's method and the related stochastic approximation theory in the following three aspects: Advancing stable law approximations via Stein's method, with explicit error bounds for systems exhibiting heavy-tailed behavior; 2) Establishing a new Stein's method for diffusion approximation, which enables a tractable steady-state analysis of queueing systems through approximations of stationary measures for stochastic differential equations. 3) Establishing a unified Markovian framework for stochastic approximations of stochastic optimization algorithms—including SGD, SVRG, and momentum-based variants—to enable comparative analysis of their convergence dynamics.
Dr . Lihu Xu received his Ph . D . from Imperial College London in 2008 and is currently an Associate Professor at the University of acau ( UM ), where he has been since 2014. Prior to joining UM , he held postdoctoral research positions at the University of Bonn , Eindhoven University of Technology , and Technische Universit ä t Berlin (2008-2011). From 2011 to 2013, he served as a Lecturer ( a permanent role equivalent to Assistant Professor ) at Brunel University London . From 09/2011 to 12/2011, Dr . Xu was a visiting scholar at Boston College . From 01/2024 to 07/2024, he was a visiting faculty in the Department of Statistics at Harvard University . Dr . Xu ' s research focuses on applied probability and its interdisciplinary applications in theoretical statistics . His work spans Stein ' s method , stochastic approximations , large deviations , Malliavin calculus , tstatisticaletimation . He has authored over 50 peer - reviewed publications in international journals such as Annals of Applied Probability , Annals of Statistics , Bernoulli , Journal of Functional Analysis , and Probability Theory and Related Fields .
