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Central Limit Theorems �and� Stein’s Method for Gaussian Approximations��(Blueprint for 2023 NCTS USRP Group 1)

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Gi-Ren Liu (National Cheng Kung University)

Yuan-Chung Sheu (National Yang Ming Chiao Tung University)

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Galton Board

Reference of the videos in this talk:

[1] https://www.youtube.com/watch?v=3m4bxse2JEQ

[2] https://www.youtube.com/watch?v=zeJD6dqJ5lo&t=574s

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A Simplified Model for the Galton Board

 

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Convergence in distribution

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Applications of CLT: Opinion Polls

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Locally independent random sequences

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2023 NCTS USRP Schedule (July 3 – August 11)�Central Limit Theorems �and� Stein’s Method for Gaussian Approximations �

  • Week 1. Central Limit Theorems [1]
  • Week 2. Stein’s Method for One-dimensional Normal Approximation [2,3]
  • Week 3. Multidimensional Stein’s Method [2,4]
  • Week 4. Non-Gaussian Approximation [4,5]
  • Week 5. Applications of Gaussian and non-Gaussian Approximation [6,7]
  • Week 6. Open Problem Discussion [6,7,8,9]

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Reference

[1] Le Gall, J. F., 2022. Measure Theory, Probability, and Stochastic Processes. Vol. 295. Springer Nature.

[2] Nourdin, I. and Peccati, G., 2012. Normal approximations with Malliavin calculus: from Stein's method to universality (Vol. 192). Cambridge University Press.

[3] Chatterjee, Sourav. "A short survey of Stein's method." arXiv preprint arXiv:1404.1392 (2014).

[4] Chen, L.H., Goldstein, L. and Shao, Q.M., 2011. Normal approximation by Stein's method (Vol. 2). Berlin: Springer.

[5] Azmoodeh, E., Eichelsbacher, P. and Thäle, C., 2022. Optimal Variance–Gamma approximation on the second Wiener chaos. Journal of Functional Analysis, 282(11), p.109450.

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[6] Hairer, M. and Shen, H., (2017). A central limit theorem for the KPZ equation. The Annals of Probability, 45(6B), pp.4167-4221.

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[7]  Liu G. R., Sheu Y. C. and Wu H. T., (2023). Central and non-central limit theorems arising from the scattering transform and its neural activation generalization. SIAM Journal on Mathematical Analysis  55(2), 1170-1213.

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[8] Azmoodeh, E., Eichelsbacher, P. and Thäle, C., (2022). Optimal Variance–Gamma approximation on the second Wiener chaos. Journal of Functional Analysis, 282(11), p.109450.

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[9] Rinott, Y. and Rotar, V. (2003). On edgeworth expansions for dependency-neighborhoods chain structures and stein’s method. Probability Theory and Related Fields, 126(4):528– 570.

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Thank You for Your Attention