Cumulants, concentration and superconcentration

• Applicant: Professorin Dr. Hanna Döring
• Subject Area: Mathematics
• Term: from 2016 to 2019
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 318196255

• Final Report Year: 2021

Final Report Abstract

The method of cumulants is a central tool in comparing the distribution of a random variable with the normal law. It enters the proof of central limit theorems, moderate and large deviation principles, Berry- Esseen bounds, and concentration inequalities in various fields of probability theory: stochastic geometry, random matrices, random graphs, random combinatorial structures, functionals of stochastic processes, mathematical biology and the list is not exhaustive. The focus of the scientific network cumulants, con- centration and superconcentration was to work through the classical proof as well as to show new areas of application. At the end of this project, we could submit a survey that shines a spotlight on a celebrated series of bounds developed by the Lithuanian school, notably Bentkus, Rudzkis, Saulis, and Statulevičius. The bounds work under a condition on cumulants that allows for heavy-tailed behavior and is considerably weaker than the Cramér condition of finite exponential moments frequently invoked in the theory of large deviations. The conditions on cumulants can be verified in many situations of interest, beyond sums of independent random variables. Great progress has been made especially in the field of stochastic geometry as a series of articles by Chri- stoph Thäle and Zakhar Kabluchko and co-authors shows. For example, they study the limit behaviour of the f-vector of random high-dimensional cones, prove central limit theorems for random simplicial complexes and study intrinsic volumes of random convex polytopes. The scientific network will continue to exist beyond the funding period for further scientific exchange among each other.

Publications

• Expected intrinsic volumes and facet numbers of random beta-polytopes, Math. Nachr. 292:1 (2017), 79–105
  Kabluchko, Zakhar; Temesvari, Daniel & Thäle, Christoph
  
• Cluster expansions for Gibbs point processes, Advances in Applied Probability 51:4 (2018), 1129–1178
  Jansen, S.
  
• Cones generated by random points on half-spheres and convex hulls of Poisson point processes, Probab. Theory Relat. Fields (2019)
  Kabluchko, Zakhar; Marynych, Alexander; Temesvari, Daniel & Thäle, Christoph
  
• High-dimensional limit theorems for p random vectors in ℓnp -balls, II, Communications in Contemporary Mathematics 23 (2021), no. 03, article 1950073
  Kabluchko, Zakhar; Prochno, Joscha & Thäle, Christoph
  
• Limit theorems for random simplices in high dimensions, ALEA, Lat. Am. J. Probab. Math. Stat.16 (2019), 141–177
  Grote, Julian; Kabluchko, Zakhar & Thäle, Christoph
  
• Beta polytopes and Poisson polyhedra: f-vectors and angles, Advances in Mathematics 374 (2020), article 107333
  Kabluchko, Zakhar; Thäle, Christoph & Zaporozhets, Dmitry
  
• Intersection of unit balls in classical matrix ensembles, Isr. J. Math. 239 (2020), 129–172
  Kabluchko, Zakhar; Prochno, Joscha & Thäle, Christoph
  
• Sanov-type large deviations in Schatten classes, Annales de l’Institut Henri Poincaré, Probabilité et Statistique 56 (2020), 928–953
  Kabluchko, Zakhar; Prochno, Joscha & Thäle, Christoph
  
• The method of cumulants for the normal approximation
  Hanna Döring, Sabine Jansen, Kristina Schubert