Kumulanten

Cumulants, Concentration and Superconcentration

The scientific network cumulants, concentration and superconcentration was funded by DFG (Project number: 318196255) from 2016 to 2019 and included workshops at the research locations Berlin, Bochum, Münster and Osnabrück:  https://gepris.dfg.de/project/318196255?lang=en. Members of the scientific network are Alessandra Cipriani, Hanna Döring, Sabine Jansen, Zakhar Kabluchko, Noemi Kurt, Marcel Ortgiese, Mathias Rafler, Kristina Schubert and Christoph Thäle.

The focus of this network 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, like in stochastic geometry for the f-vector of random high-dimensional cones, for random simplicial complexes and intrinsic volumes of random convex polytopes.

Literature: Method of Cumulants

This list below aims to establish an extensive overview of the existing literature concerning the method of cumulants which is a technique to prove convergence in distribution. A particularly useful technique of this method is the Statulevičius condition, which provides quantitative results for the convergence in distribution. We do not claim, that this list is exhaustive. If the reader might find an article that fits the list and is missing, please contact the editor of this page (fabian.apostel@uni-osnabrueck.de).

The list is structured according to the following categories:

Foundations and general references on cumulants

  1. V. Leonov and A. Shiryaev (1959):  On a Method of Calculation of Semi-Invariants
  2. D. R. Brillinger (1969):  The calculation of cumulants via conditioning
  3. T. Speed (1983):  Cumulants and partition lattices
  4. P. McCullagh (1987):  Tensor methods in statistics.
  5. O. Barndorff-Nielsen and D. Cox (1989):  Asymptotic techniques for use in statistics.
  6. W. Bryc (1993):  A remark on the connection between the large deviation principle and the central limit theorem
  7. G.-C. Rota and J. Shen (2000):  On the Combinatorics of Cumulants
  8. A. Hald (2007):  The Early History of the Cumulants and the Gram-Charlier Series (Historical overview)
  9. B. Sturmfels and P. Zwiernik (2013):  Binary Cumulant Varieties
  10. O. Arizmendi, T. Hasebe, F. Lehner and C. Vargas (2015):  Relations between cumulants in noncommutative probability
  11. P. Zwiernik (2016):  Semialgebraic statistics and latent tree models.
  12. J. Mingo and R. Speicher (2017):  Free Probability and Random Matrices
  13. H. Cramér and H. Touchette (translator) (2018):  On a new limit theorem in probability theory (Translation of 'Sur un nouveau théorème-limite de la théorie des probabilités')
  14. J. Lukkarinen, M Marcozzi and A. Nota (2018):  Summability of Connected Correlation Functions of Coupled Lattice Fields
  15. T. Bodineau, I. Gallagher, L. Saint-Raymond and S. Simonella (2023):  Statistical dynamics of a hard sphere gas: fluctuating Boltzmann equation and large deviations

Methods for obtaining cumulant bounds

  1. J. L. Lebowitz (1972):  Bounds on the correlations and analyticity properties of ferromagnetic Ising spin systems
  2. M. Duneau, D. Iagolnitzer and B. Souillard (1973):  Decrease properties of truncated correlation functions and analyticity properties for classical lattices and continuous systems
  3. D.R. Brillinger (1975): Statistical Inference for Stationary Point processes
  4. L. Heinrich and W.-D. Richter (1984):  On Moderate Deviations of Sums of m-Dependent Random Vectors
  5. L. Heinrich (1985):  Some Estimates of the Cumulant-Generating Function of a Sum of m-Dependent Random Vectors and Their Application to Large Deviations
  6. L. Heinrich (1990):  Some Bounds of Cumulants of m-Dependent Random Fields
  7. F. Götze, L. Heinrich and C. Hipp (1995):  m-Dependent Random Fields with Analytic Cumulant Generating Function
  8. R. S. Kallabis and M. H. Neumann (2006):  An exponential inequality under weak dependence
  9. P. Doukhan and M. H. Neumann (2007):  Probability and moment inequalities for sums of weakly dependent random variables, with applications
  10. I. Nourdin and G. Peccati (2010):  Cumulants on the Wiener space
  11. G. Peccati and M. Taqqu (2011):  Wiener Chaos: Moments, Cumulants and Diagrams
  12. G. Last, M. D. Penrose, M. Schulte and C. Thäle (2016):  Moments and Central Limit Theorems for Some Multivariate Poisson Functionals
  13. V. Féray , P.-L. Méliot and A. Nikeghbali (2016):  Mod-ϕ Convergence, Normality Zones and Precise Deviations
  14. V. Féray (2018):  Weighted dependency graphs
  15. V. Féray , P.-L. Méliot and A. Nikeghbali (2019):  Mod-ϕ Convergence, II: Estimates on the Speed of Convergence
  16. N. Heerten, H. Sambale and C. Thäle (2023):  Probabilistic limit theorems induced by the zeros of polynomials
  17. M. Michelen and J. Sahasrabudhe (2024):  Central limit theorems and the geometry of polynomials
  18. M. Janisch and T. Lehéricy (2024):  Berry–Esseen-Type Estimates for Random Variables with a Sparse Dependency Graph
  19. A. Durmus, E. Moulines, A. Naumov and S. Samsonov (2024):  Probability and Moment Inequalities for Additive Functionals of Geometrically Ergodic Markov Chains
  20. M. Isaev (2025):  A tail bound for cumulant series for complex functions of independent random variables
  21. J. Zhang (2025): Explicit Universal Bounds for Cumulants via Moments

Cumulant method as a method for proving qualitative limit theorems

General theory:

  1. S. Janson (1988):  Normal Convergence by Higher Semiinvariants with Applications to Sums of Dependent Random Variables and Random Graphs
  2. M. Michelen and J. Sahasrabudhe (2019):  Central limit theorems from the roots of probability generating functions

Random matrices and determinantal processes:

  1. O. Costin and J. Lebowitz (1995):  Gaussian Fluctuation in Random Matrices
  2. A. Soshnikov (1998):  Level Spacings Distribution for Large Random Matrices: Gaussian Fluctuations
  3. A. Soshnikov (2000):  The central limit theorem for local linear statistics in classical compact groups and related combinatorial identities
  4. J. P. Keating and N. C. Snaith (2000):  Random Matrix Theory and ζ(1/2+it)
  5. A. Soshnikov (2002):  Gaussian Limit for Determinantal Random Point Fields
  6. B. Rider and B. Virag (2006):  The noise in the circular law and the Gaussian free field
  7. Y. Ameur, H. Hedenmalm and N. Makarov (2011):  Fluctuations of eigenvalues of random normal matrices
  8. G. Borot and A. Guionnet (2013):  Asymptotic Expansion of β Matrix Models in the One-cut Regime
  9. R. Chhaibi, J. Najnudel and A. Nikeghbali (2016):  The circular unitary ensemble and the Riemann zeta function: the microscopic landscape and a new approach to ratios
  10. G. Pan, S. Wang and W. Zhou (2017):  Limit theorems for linear spectrum statistics of orthogonal polynomial ensembles and their applications in random matrix theory
  11. G. Lambert (2018):  Limit theorems for biorthogonal ensembles and related combinatorial identities
  12. G. Lambert (2018):  Mesoscopic fluctuations for unitary invariant ensembles
  13. H. Dette and D. Tomecki (2019):  Determinants of block Hankel matrices for random matrix-valued measures
  14. M. Fenzl and G. Lambert (2021):  Precise Deviations for Disk Counting Statistics of Invariant Determinantal Processes
  15. R. Feng, F. Götze and D. Yao (2023):  Determinantal point processes on spheres: multivariate linear statistics
  16. A. Soshnikov and C. Wu (2023):  A Note on Cumulant Technique in Random Matrix Theory
  17. S.-S. Byun, L. Molag and N. Simm (2025):  Large deviations and fluctuations of real eigenvalues of elliptic random matrices
  18. H. Zhao (2026):   Sharp Berry–Esseen Bounds for the Log Determinant of a Gaussian Sample Correlation Matrix

Stochastic geometry and point processes:

  1. D. R. Brillinger and M. Rosenblatt (1966): Asymptotic Theory of Estimates of kTH-Order Spectra
  2. D.R. Brillinger (1969):  Asymptotic Properties of Spectral Estimates of Second Order
  3. D.R. Brillinger (1974): Fourier analysis of stationary processes
  4. D.R. Brillinger (1975): Statistical Inference for Stationary Point processes
  5. D. R. Brillinger (1977):  The identification of a particular nonlinear time series system
  6. G. Ivanoff (1982):  Central limit theorems for point processes
  7. D.R. Brillinger (1982):  Asymptotic normality of finite Fourier transforms of stationary generalized processes
  8. Y. Baryshnikov and J. Yukich (2005):  Gaussian limits for random measures in geometric probability
  9. B. Błaszczyszyn, D. Yogeshwaran and J. Yukich (2019):  Limit theory for geometric statistics of point processes having fast decay of correlations
  10. L. Heinrich and M. Spiess (2016):  Central Limit Theorems for Volume and Surface Content of Stationary Poisson Cylinder Processes in Expanding Domains
  11. N. Privault (2024):  Asymptotic analysis of k-hop connectivity in the 1D unit disk random graph model
  12. J. Jalowy and H. Stange (2025):  Box-Covariances of Hyperuniform Point Processes
  13. B. Błaszczyszyn, D. Yogeshwaran and J. Yukich (2026):  Limit theory for Lipschitz-localized statistics in random geometric models
  14. G. Mastrilli (2026):  Asymptotic fluctuations of smooth linear statistics of independently perturbed lattices

Statistical mechanics and interacting system

  1. D. Iagolnitzer and B. Souillard (1978):  Lee-Yang theory and normal fluctuations
  2. J. Dousse and V. Féray (2019):  Weighted dependency graphs and the Ising model
  3. T. Bodineau, I. Gallagher, L. Saint-Raymond and S. Simonella (2023):  Statistical dynamics of a hard sphere gas: fluctuating Boltzmann equation and large deviations
  4. A. C. Benn (2025):  A Berry-Esseen theorem for wide quantum neural networks
  5. F. Girardi and G. De Palma (2025):  Trained Quantum Neural Networks are Gaussian Processes 
  6. P. Śniady (2026):  Pfaffian structure of basin walls for coalescing particles

Random graphs, combinatorics, permutations and coalescents

  1. M. Moehle and H. Pitters (2015):  Absorption Time and Tree Length of the Kingman Coalescent and the Gumbel Distribution
  2. H. Pitters (2017):  On the number of segregating sites
  3. V. Féray (2020): Central limit theorems for patterns in multiset permutations and set partitions
  4. V. Féray, P.-L. Méliot and A. Nikeghbali (2020):  Graphons, permutons and the Thoma simplex: three mod-Gaussian moduli spaces
  5. V. Féray and M. S. Kammoun (2024):  Asymptotic normality of pattern counts in conjugacy classes
  6. R. S. Hazra, N. Kriukov, M. Mandjes and M. Otto (2025):  Functional central limit theorem for subgraph counts in a dynamic random connection model

Dynamical systems and number theory

  1. S. Janson (2013):  Euler-Frobenius numbers and rounding
  2. M. Björklund and A. Gorodnik (2019):  Central limit theorems for Diophantine approximants
  3. M. Björklund and A. Gorodnik (2020):  Central limit theorems for group actions which are exponentially mixing of all orders
  4. S. Das (2024):  Central limit theorems for lattice point counting on tessellated domains
  5. G. Aggarwal and Anish Ghosh (2025):  Two central limit theorems in Diophantine approximation
  6. S. Li (2026):  The invariance principle for inhomogeneous Diophantine approximations
  7. G. Aggarwal, S. Das and A. Ghosh (2026):  Effective multi-equidistribution for translates of unipotent flows and Central limit theorems in inhomogeneous Diophantine approximation

Gaussian functionals

  1. L. Neufcourt and F. Viens (2013):  A third-moment theorem and precise asymptotics for variations of stationary Gaussian sequences
  2. L. Gass (2023):  Cumulants asymptotics for the zeros counting measure of real Gaussian processes
  3. H. Chen (2024):  Optimal Rate of Convergence for Vector-valued Wiener-Ito Integral
  4. M. Ancona, L. Gass, T. Letendre and M. Stecconi (2025):  Zeros and critical points of Gaussian fields: cumulants asymptotics and limit theorems

Statulevičius condition for proving quantitative limit theorems

General theory

  1. R. Fisher and J. Wishart (1932):  The Derivation of the Pattern Formulae of Two-Way Partitions from those of Simpler Patterns
  2. V. Statuljavičius (1966):  On large deviations
  3. R. Rudzkis, L. Saulis and V. Statuljavičus (1978):  A general lemma on probabilities of large deviations. Moreover,  here you can find a link, to a pdf-version at Springer.
  4. L. Saulis (1979):  Large deviations for sums of independent weighted random variables
  5. V. Statulevičius (1979):  Large-deviation theorems for sums of dependent random variables. I
  6. R. Lapinskas (1980):  Limit theorems for weakly dependent random variables
  7. R. Bentkus and R. Rudzkis (1980):  Exponential estimates for the distribution of random variables.
  8. A. Plikusas (1980):  Estimation of cumulants and large deviations for certain nonlinear transformations of a stationary Gaussian process
  9. R. Bentkus and R. Rudzkis (1983):  On the distribution of some statistical estimates of spectral density
  10. L. Heinrich and W.-D. Richter (1984):  On Moderate Deviations of Sums of m-Dependent Random Vectors
  11. L. Heinrich (1985):  Some Estimates of the Cumulant-Generating Function of a Sum of m-Dependent Random Vectors and Their Application to Large Deviations
  12. S. Čepulenas (1985):  Probabilities of large deviations for random fields
  13. L. Heinrich (1986):  A method for the derivation of limit theorems for sums of weakly dependent random variables:a survey
  14. L. Saulis (1987):  General lemmas on large deviations for a random vector with regular behavior of cumulants. I
  15. V. Statuljavičius and D. Jakimavičius (1988):  Estimates of semiinvariants and centered moments of stochastic processes with mixing. I
  16. D. Jakimavičius (1988):  Estimates of cumulants and centered moments of mixing stochastic processes
  17. L. Saulis and V. Statuljavičus (1989; Translation in (1991) :  Предельные теоремы о больших уклонениях. The translation was published in 1991:  Limit Theorems for Large Deviations.
  18. S. Achmedov (1990):  Nonuniform estimates in the central limit theorem for dependent variables
  19. V. Bentkus and R. Zitikis (1990):  Probabilities of large deviations for L-statistics
  20. A. K. Aleshkevichiene (1991):  Probabilities of Large Deviations for U-Statistics and von Mises Functionals
  21. D.R. Brillinger (1992):  Moments, cumulants and some applications to stationary random processes
  22. L. Saulis (1992):  Probabilities of large deviations for random vectors
  23. S. Čepulenas (1993):  Large deviations for additive functionals of d-dependent random fields.
  24. R. Rudzkis (1993):  On the distribution of supremum-type functionals of nonparametric estimates of probability and spectral densities
  25. V. Statuljavičis and A. Aleshkevichiene (1994):  On Large Deviations in the Poisson Approximation
  26. A. K. Aleshkevichiene and V. Statuljavičis (1994): Large deviations in approximation by Poisson law
  27. L. Saulis (1995):  Asymptotic expansion of the distribution function of a random variable with regular behavior of cumulants
  28. F. Götze, L. Heinrich and C. Hipp (1995):  m-Dependent Random Fields with Analytic Cumulant Generating Function
  29. A. B. Gorchakov (1995):  Upper estimates for semi-invariants of the sum of multi-indexed random variables
  30. A. Aleshkevichiene and V. Statuljavičius (1997):  Probabilities of large deviations in the approximation by χ2-law 
  31. P. Vaitkus and V. Čekanavičius (1998):  On a centered Poisson approximation
  32. K. Padvelskis and V. Statulevičius (1998):  Theorems of large deviations for sums of random variables connected in a Markov chain. I
  33. V. Čekanavičius and P. Vaitkus (1999):  Large deviations for integer centered poisson approximation
  34. L. Saulis (1999):  Asymptotic Expansions of Large Deviations for Sums of Nonidentically Distributed Random Variables
  35. K. Padvelskis and V. Statulevičius (1999):  Theorems of large deviations for sums of random variables connected in a Markov chain. II
  36. N. Amosova (1999):  The necessity of the Linnik condition in a theorem on probabilities of large deviations
  37. N. Amosova (1999):  On the necessity of Statulevičius' condition in limit theorems for large-deviation probabilities
  38. N. Amosova (2002):  Necessity of the Cramer, Linnik, and Statulevičius Conditions for Large-Deviation Probabilities
  39. S. Čepulenas (2003):  On Large Deviations of Vector-Valued Additive Functions of a Markov Gaussian Field
  40. L. Saulis and D. Deltuviene (2007):  Normal approximation for sum of random number of summands
  41. A. Kasparavičiute and L. Saulis (2010):  Theorems on large deviations for the sum of random number of summands
  42. A. Kasparavičiute and L. Saulis (2011):  Theorems on Large Deviations for Randomly Indexed Sum of Weighted Random Variables
  43. A. Kasparavičiute and L. Saulis (2013):  Large deviations for weighted random sums
  44. H. Döring and P. Eichelsbacher (2013):  Moderate Deviations via Cumulants
  45. A. Kasparavičiute and D. Deltuviene (2015):  On large deviations for random sums of the squares of weighted Gaussian random variables
  46. R. Rudzkis and A. Bakshaev (2017):  General theorems on large deviations for random vectors
  47. P. Eichelsbacher and L. Knichel (2019):  Moment estimates of Rosenthal type via cumulants
  48. P. Eichelsbacher and M. Löwe (2019):  Lindeberg’s Method for Moderate Deviations and Random Summation
  49. H. Döring, S. Jansen and K. Schubert (2022):  The method of cumulants for the normal approximation
  50. A. Y. Zaitsev (2025):  Estimates of transport distance in the central limit theorem

Random matrices and determinantal processes

  1. L. Heinrich (2016):  On the strong Brillinger-mixing property of α-determinantal point processes and some applications
  2. R. Feng, F. Götze and  D. Yao (2023):  Determinantal point processes on spheres: multivariate linear statistic
  3. Z. Bao and D. M. George (2025): Ultra high order cumulants and quantitative CLT for polynomials in random matrices

Stochastic geometry and point processes:

  1. A. Plikusas (1995): Limit theorems for some polynomial statistics of the poisson process
  2. L. Heinrich (2005):  Large deviations of the empirical volume fraction for stationary Poisson grain models
  3. L. Heinrich (2007):  An Almost-Markov-Type Mixing Condition and Large Deviations for Boolean Models on the Line
  4. L. Heinrich and M. Spiess (2009):  Berry–Esseen bounds and Cramér-type large deviations for the volume distribution of Poisson cylinder processes
  5. P. Eichelsbacher, M. Raič and T. Schreiber (2015):  Moderate deviations for stabilizing functionals in geometric probability
  6. J. Grote and C. Thäle (2018):  Concentration and moderate deviations for Poisson polytopes and polyhedra
  7. J. Grote and C. Thäle (2018):  Gaussian polytopes: A cumulant-based approach
  8. J. Grote, Z. Kabluchko and C. Thäle (2019):  Limit theorems for random simplices in high dimensions
  9. M. Fenzl (2019):  Asymptotic results for stabilizing functionals of point processes having fast decay of correlations
  10. A. Gusakova and C. Thäle (2021):  On random convex chains, orthogonal polynomials, PF sequences and probabilistic limit theorems
  11. A. Gusakova and C. Thäle (2021):  The volume of simplices in high-dimensional Poisson–Delaunay tessellations
  12. A. Gusakova, Z. Kabluchko and C. Thäle (2022):  The β-Delaunay tessellation III: Kendall’s problem and limit theorems in high dimensions
  13. N. Heerten (2024):  The cumulant method for counting variables and typical cells in poisson line tessellations (Phd Thesis)

Statistical mechanics and interacting particle systems

  1. P.-L. Méliot and A. Nikeghbali (2015):  Mod-Gaussian convergence and its applications for models of statistical mechanics.
  2.  G. Giacomin, A. Legrand and M. Zamparo (2026):  The localized phase of pinning models with correlated Gaussian disorder
  3.  F. Apostel, H. Döring and K. Schubert (2026):  Fine asymptotics of the magnetization of the annealed dilute Curie-Weiss model
  4.  G. Giacomin and M. Zamparo (2024):  Concentration and fluctuation phenomena in the localized phase of the pinning model
  5. G. Mastrilli (2025): Minimax estimation of the structure factor of spatial point processes 
  6. P. Xie (2025):  Uniform-in-Time Estimates on the Size of Chaos for Interacting Particle Systems

Random graphs, combinatorics, permutations and coalescents

  1. I. Narang, W. Perkins and T. L. H. Wee (2026):  Optimal detection of planted stars via a random energy model
  2. L. Hofer (2017):  A Central Limit Theorem for Vincular Permutation Patterns
  3. Q. Liu and N. Privault (2023):  Normal approximation of subgraph counts in the random-connection model
  4. Q. Liu and N. Privault (2025): Normal to Poisson phase transition for subgraph counting in the random-connection model
  5. N. Heerten, C. Hirsch and M. Otto (2025):  Moderate deviations for weight-dependent random connection models
  6. Q. Liu and N. Privault (2025):  Graph connectivity with fixed endpoints in the random-connection model
  7. Q. Liu and N. Privault (2025):  Gaussian fluctuations of generalized U-statistics and subgraph counting in the binomial random-connection model

Dynamical systems and number theory

  1. B. Kryžiene and G. Misevičius (2008):  On limit theorems of Fortet–Kac type
  2. P. Kamieński (2019):  A quantitative version of the theorem on Khintchine’s constant
  3. J. Prochno and M. Strzelecka (2023): Moderate deviation principles and Mod-Gaussian convergence for lacunary trigonometric sums
  4. K. Holm (2023):  A Central Limit Theorem for Counting Functions Related to Symplectic Lattices and Bounded Sets
  5. Y. Hafouta (2023):  Convergence rates in the functional CLT for α-mixing triangular arrays
  6. Y. Hafouta (2024):  Large deviations, moment estimates and almost sure invariance principles for skew products with mixing base maps and expanding-on-average fibers

Times series and statistical estimation

  1. M. H. Neumann (1996):  Spectral density estimation via nonlinear wavelet methods for stationary non-Gaussian time series
  2. M. H. Neumann and R. von Sachs (1997):  Wavelet thresholding in anisotropic function classes and application to adaptive estimation of evolutionary spectra
  3. R. Dahlhaus, M. H. Neumann and R. von Sachs (1999):  Nonlinear wavelet estimation of time-varying autoregressive processes
  4. R. Dahlhaus and M. H. Neumann (2001):  Locally adaptive fitting of semiparametric models to nonstationary time series
  5. R. von Sachs and Brenda Macgibbon (2002):  Non-parametric Curve Estimation by Wavelet Thresholding with Locally Stationary Errors
  6. S. Touati and J.-C. Pesquet (2002):  Wavelet-thresholding for bispectrum estimation
  7. S. Touati and J.-C. Pesquet (2003):  Wavelet estimation of cyclospectra
  8. D. R. Brillinger (2007):  Some uses if cumulants in wavelet analysis
  9. J. Chau and R. von Sachs (2016):  Functional mixed effects wavelet estimation for spectra of replicated time series

Gaussian and Poisson functionals

  1. M. Schulte and C. Thäle (2016):  Cumulants on Wiener chaos: Moderate deviations and the fourth moment theorem
  2. M. Khabou and G. L. Torrisi (2024):  Gaussian approximation and moderate deviations of Poisson shot noises with application to compound generalized Hawkes processes
  3. M. Schulte and C. Thäle (2024):  Moderate deviations on Poisson chaos

Workshops

opening workshop on Cumulants, Concentration and Superconcentration
6th to 8th December 2016

We are glad that two experts of this field,  Lothar Heinrich from Augsburg and  Valentin Féray from Zurich, accepted our invitation and gave two mini courses.

Gruppenbild

The opening workshop took place at the Institute for Mathematics, Albrechtstraße 28 a, 49076 Osnabrück. (You can find directions  here.)

Programme:

Tuesday
9:30 a.m. Welcome and Coffee
10:00 a.m. Opening Meeting of the members of the network
12:30 p.m. Lunch
14:00 p.m Valentin Féray, mod-φ-convergence I
Coffee Break
15:30 p.m. Work in Groups

Wednesday
9:00 a.m. Work in Groups
11:00 a.m. Coffee Break
11:30 a.m. Valentin Féray, mod-φ-convergence II
Lunch
14:00 p.m. Work in Groups
16:45 p.m. Colloquium Coffee Break
17:00 p.m. Colloquium Talk by Anja Sturm On Theory and Applications of
Interacting Particle Systems
18:30 p.m. Dinner in Italian Job

Thursday
9:00 a.m. Lothar Heinrich, Cumulants and Their Role in Proving Central Limit Theorems
Coffee Break
10:30 a.m. Work in Groups
11:30 p.m. Lothar Heinrich, On the Brillinger-Mixing Property of Stationary Point Processes
Lunch
14:00 p.m. Work in Groups
15:00 p.m. Status quo and Feedback

The second workshop of the DFG scientific network will take place 26th and 27th June 2017 at Ruhr-Universität Bochum

SPEAKERS:
 Peter Eichelsbacher, Bochum
 Maurizia Rossi, Luxembourg

VENUE:  Ruhr-Universität Bochum, building NA, see  campus map

PROGRAMME:

Programme

The third workshop of the scientific network cumulants, concentration and superconcentration will take place on 5th and 6th April 2018 at TU Berlin

We are looking forward to the mini courses by  Anton Klimovsky and  Helmut Pitters.

PROGRAMME:

Thursday:

10:00 — 12:00 Arrival and group work, in MA 748 (and MA 841)

12:00 — 14:00 Lunch break

14:00 — 15:00 (Part 1) Helmut Pitters, in MA 748

15:00 — 16:00 Group work and coffee

16:00 –17:00 (Part 2) Helmut Pitters, in MA 748

17:00 Group work
Friday:

09:30 –10:30 High-dimensional Gaussian fields seen through spin glasses, Anton Klimovsky, in MA 748

10:30 — 11:30 coffee break

11:30 — 12:30 Complex random energy models, Anton Klimovsky, in MA 748

12:30 — 14:00 Lunch break

14:00 — 16:00 Group work and coffee

The fourth workshop of the scientific network cumulants, concentration and superconcentration will take place on 10 and 11 January 2019 at Westfälische Wilhelms-Universität Münster

We are looking forward to courses by our guests  Bartek Błaszczyszyn and  Matthias Schulte.

Evereybody is welcome to work in small groups on Wednesday. The  programme (PDF, 27 kB) will start on Thursday, January 10th:
10:00 –12:00   arrival and group work
12:00 – 14:00  lunch break
14:00 – 15:00  talk by Bartek Błaszczyszyn
15:00 – 16:00  group work and coffee
16:00 – 17:00  talk by Matthias Schulte
17:00 – 18:30  group work
19:00 dinner

Friday, January 11th
09:00 – 10:00 talk by Bartek Błaszczyszyn
10:00 – 11:00   group work and coffee
11:00 – 12:00   talk by Matthias Schulte
12:00 – 13:30   lunch break
13:30 – 16:00  group work and coffee

The workshop  poster (PDF, 2,57 MB) can be downloaded here.

Venue:

University of Münster

Institute for Mathematical Stochastics,

Orléans-Ring 10, 48149 Münster

Room:

N2, groundfloor of Orléans-Ring 10

The fifth workshop of the scientific network cumulants, concentration and superconcentration will take place on September 31st  to October 2nd 2019  at University of Osnabrück.

We are looking forward to courses by our guests Christian Hirsch, Ivan Nourdin & Giovanni Peccati.

Program
Monday, 30.9.
all day                Welcome and time to work
13:00 – 14:00  members of the scientific network: “Abschlussbericht und Zukunftsperspektiven”
14:30 – 15:30  Functional CLT for persistent Betti numbers on point processes 1 by Christian Hirsch

Tuesday, 1.10.
09:30 – 10:30   Functional CLT for persistent Betti numbers on point processes 2 by Christian Hirsch
10:30 – 11:30   group work and coffee
11:30 – 12:30   Limit theorems for functionals of random measures: combinatorial aspects and integration by parts 1 by Giovanni Peccati
12:30 – 14:00   lunch break
14:00 – 15:00   Limit theorems for functionals of random measures: combinatorial aspects and integration by parts 2 by  Giovanni Peccati
15:00 – 15:30   coffee break
15:30 – 16:30   Around the Breuer-Major theorem 1 by Ivan Nourdin
16:30 – 17:30   group work

Wednesday, 2.10.
09:30 – 10:30  Around the Breuer-Major theorem 2 by Ivan Nourdin
10:30 – 12:00   group work and coffee
12:00 – 12:30   The volume of simplices in high-dimensional Poisson-Delaunay tessellations by Anna Gusakova
12:30 – 14:00   lunch break
14:00 – 16:00   group work and coffee

The workshop poster can be downloaded  here. (PDF, 2,57 MB)

Venue:
University of Osnabrück
Institute for Mathematics,
Albrechtstraße 28a, 49076 Osnabrück
Building No. 69
All talks will take place in the first floor, room 69/125.

Abstracts

Christian Hirsch: Functional CLT for persistent Betti numbers on point processes
The persistence diagram and persistent Betti numbers form one of the key tools in topological data analysis, an emerging methodology to leverage invariants from algebraic topology to extract insights from data. The two lectures are devoted to proving a functional CLT for the persistence diagram on point processes via cumulative methods. In the first lecture, we discuss in detail the role of persistent Betti Numbers in topological data analysis. We start with the intuition, give the precise definition in terms of homology of simplicial complexes and conclude with a selection of current computation algorithms. The second lecture describes how to approach the CLT by combining the Bickel-Wichura machinery with cumulant measures associated with point processes. This program works in the the planar case and for bounded features. We conclude by discussing challenges in higher dimensions and possibly unbounded features. This talk is based on joint work with Christophe Biscio, Nicolas Chenavier and Anne Marie Svane.

Ivan Noudin: Around the Breuer-Major theorem
Let X be a centered, unit-variance and stationary Gaussian sequence with covariance function r. Let f be a square integrable function with respect to the standard Gaussian measure on the real line. Consider the sequence F_n of partial sums associated with f(X_n). The celebrated Breuer-Major theorem provides sufficient conditions on the covariance r and of the Hermite index of f in order for F_n to exhibit Gaussian fluctuations. It has far-reaching applications in many different areas, such as mathematical statistics, signal processing or geometry of random nodal sets.
The goal of this series of of two talks is to introduce the audience to this beautiful result. We will then review some recent applications and extensions. All the needed material will be introduced progressively; no specific knowledge is required, beyond the definitions and very basic properties of Gaussian processes (mostly Brownian motion).

Giovanni Peccati: Limit theorems for functionals of random measures: combinatorial aspects and integration by parts
I will explore some parts of a recent body of work, focussing on limit theorems for (Gaussian or point) random measures obtained by using methods of infinite-dimensional stochastic analysis, such as Malliavin calculus and integration by parts formulae. I will try to highlight some of the combinatorial implications of our main results (mostly related to the so-called Rota-Wallstrom Theory), and also to develop and illustrate some simple geometric applications.

Publications

Sabine Jansen, Cluster expansions for GIBBS point processes, Advances in Applied Probability 51:4 (2019), 1129-1178,  https://doi.org/10.1017/apr.2019.46.

Nicholas M. Ercolani, Sabine Jansen & Daniel Ueltschi, Singularity analysis for heavy-tailed variables, J Theor. Probab. (2018),  https://doi.org/10.1007/s10959-018-0832-2

Zakhar Kabluchko, Alexander Marynych, Daniel Temesvari, Christoph Thäle, Cones generated by random points on half-spheres and convex hulls of Poisson point processes, Probab. Theory Relat. Fields (2019),  https://doi.org/10.1007/s00440-019-00907-3

Julian Grote, Zakhar Kabluchko, Christoph Thäle, Limit theorems for random simplices in high dimensions, ALEA, Lat. Am. J. Probab. Math. Stat.16 (2019),  https://doi.org/10.30757/ALEA.v16-06, 141–177.

Zakhar Kabluchko, Daniel Temesvari, Christoph Thäle, Expected intrinsic volumes and facet numbers of random beta-polytopes, 2017,   https://doi.org/10.1002/mana.201700255

Zakhar Kabluchko, Joscha Prochno, Christoph Thäle, Intersection of unit balls in classical matrix ensembles, Isr. J. Math. 239, 129–172 (2020).  https://doi.org/10.1007/s11856-020-2052-6

Zakhar Kabluchko, Joscha Prochno, Christoph Thäle, Exact asymptotic volume and volume ratio of Schatten unit balls,  Journal of Approximation Theory 257, Article 105457 (2020)

Zakhar Kabluchko, Christoph Thäle, Dmitry Zaporozhets, Beta polytopes and Poisson polyhedra: f-vectors and angles,  Advances in Mathematics 374, article 107333 (2020)

Zakhar Kabluchko, Joscha Prochno, Christoph Thäle, Sanov-type large deviations in Schatten classes,  Annales de l’Institut Henri Poincaré, Probabilité et Statistique 56:2, 928–953 (2020)

Zakhar Kabluchko, Joscha Prochno, Christoph Thäle, High-dimensional limit theorems for random vectors in ℓ_n^p-balls. II, Communications in Contemporary Mathematics (2019+),  https://doi.org/10.1142/S0219199719500731

Anna Gusakova, Christoph Thäle, The volume of simplices in high-dimensional Poisson-Delaunay tessellations, preprint,  https://arxiv.org/abs/1909.05589

Anastas Baci, Carina Betken, Anna Gusakova, Christoph Thäle, Concentration inequalities for functionals of Poisson cylinder processes,  Electronic Journal of Probability, Article 128 (2020)

Hanna Döring, Sabine Jansen, Kristina Schubert, The method of cumulants for the normal
approximation, preprint,  http://arxiv.org/abs/2102.01459