Contemporary Problems in Probability and Statistics: Gaussian Approximations and Small Deviations for Stochastic Processes

• Applicant: Professor Dr. Friedrich Götze
• Subject Area: Mathematics
• Term: from 2017 to 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 323605340

• Final Report Year: 2023

Final Report Abstract

The cooperation partners have studied small deviations for classes of stochastic processes and Gaussian approximations in probability. They studied breakpoints of Brownian chains subject to random and deterministic interactions, persistence problems for fractional Brownia motions and developed formulas for sums of Grassmann angles in stochastic geometry extending results of Sudakov and Tsireson. Furthermore optimal concentration bounds were derived for the multivariate and infinite dimensional approximation of sums of independent random vectors by infinite divisible distributions related to the second uniform limit theorem of Kolmogorov.

Publications

• New applications of Arak’s inequalities to the Littlewood–Offord problem. European Journal of Mathematics, 4(2), 639-663.
  Götze, Friedrich & Zaitsev, Andrei Yu.
  
• The First Exit Time of Fractional Brownian Motion from a Parabolic Domain. Theory of Probability & Its Applications, 64(3), 490-497.
  Aurzada, F. & Lifshits, M. A.
  
• An Improved Multivariate Version of Kolmogorov’s Second Uniform Limit Theorem. Journal of Mathematical Sciences, 258(6), 782-792.
  Götze, F.; Zaitsev, A. Yu. & Zaporozhets, D.
  
• Breaking a chain of interacting Brownian particles. The Annals of Applied Probability, 31(6).
  Aurzada, Frank; Betz, Volker & Lifshits, Mikhail
  
• Breaking a Chain of Interacting Brownian Particles: A Gumbel Limit Theorem. Theory of Probability & Its Applications, 66(2), 184-208.
  Aurzada, F.; Betz, V. & Lifshits, M.
  
• Universal break law for a class of models of polymer rupture. Journal of Physics A: Mathematical and Theoretical, 54(30), 305204.
  Aurzada, Frank; Betz, Volker & Lifshits, Mikhail
  
• On Alternative Approximating Distributions in the Multivariate Version of Kolmogorov's Second Uniform Limit Theorem. Theory of Probability & Its Applications, 67(1), 1-16.
  Götze, F. & Zaitsev, A. Yu.
  
• Angle Sums of Random Polytopes. Michigan Mathematical Journal, 73(4).
  Godland, Thomas; Kabluchko, Zakhar & Zaporozhets, Dmitry
  
• Convergence to Infinite-Dimensional Compound Poisson Distributions on Convex Polyhedra. Journal of Mathematical Sciences, 273(5), 732-737.
  Götze, F. & Zaitsev, A. Yu.
  
• Grassmann Angles and Absorption Probabilities of Gaussian Convex Hulls. Journal of Mathematical Sciences, 273(5), 738-754.
  Götze, F.; Kabluchko, Z. & Zaporozhets, D.