Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

I. Random Dynamical Systems and Their Generators.- 1. Basic Definitions. Invariant Measures.- 2. Generation.- II. Multiplicative Ergodic Theory.- 3. The Multiplicative Ergodic Theorem in Euclidean Space.- 4. The Multiplicative Ergodic Theorem on Bundles and Manifolds.- 5. The MET for Related Linear and Affine RDS.- 6. RDS on Homogeneous Spaces of the General Linear Group.- III. Smooth Random Dynamical Systems.- 7. Invariant Manifolds.- 8. Normal Forms.- 9. Bifurcation Theory.- IV. Appendices.- Appendix A. Measurable Dynamical Systems.- A.1 Ergodic Theory.- A.2 Stochastic Processes and Dynamical Systems.- A.3 Stationary Processes.- A.4 Markov Processes.- Appendix B. Smooth Dynamical Systems.- B.1 Two-Parameter Flows on a Manifold.- B.4 Autonomous Case: Dynamical Systems.- B.5 Vector Fields and Flows on Manifolds.- References.

Random Dynamical Systems

Starting around the late 1950s, several research communities began relating the geometry of graphs to stochastic processes on these graphs. This book, twenty years in the making, ties together research in the field, encompassing work on percolation, isoperimetric inequalities, eigenvalues, transition probabilities, and random walks. Written by two leading researchers, the text emphasizes intuition, while giving complete proofs and more than 850 exercises. Many recent developments, in which the authors have played a leading role, are discussed, including percolation on trees and Cayley graphs, uniform spanning forests, the mass-transport technique, and connections on random walks on graphs to embedding in Hilbert space. This state-of-the-art account of probability on networks will be indispensable for graduate students and researchers alike.

Probability on Trees and Networks

The spectral rigidity ⊿( L ) of a set of quantal energy levels is the mean square deviation of the spectral staircase from the straight line that best fits it over a range of L mean level spacings. In the semiclassical limit (ℏ→0), formulae are obtained giving ⊿( L ) as a sum over classical periodic orbits. When L ≪ L max , where L max ~ℏ-(N-1) for a system of N freedoms, ⊿( L ) is shown to display the following universal behaviour as a result of properties of very long classical orbits: if the system is classically integrable (all periodic orbits filling tori), ⊿( L )═ 1 / 5 L (as in an uncorrelated (Poisson) eigenvalue sequence); if the system is classically chaotic (all periodic orbits isolated and unstable) and has no symmetry, ⊿( L ) ═ In L /2π 2 + D if 1≪ L ≪ L max (as in the gaussian unitary ensemble of random-matrix theory); if the system is chaotic and has time-reversal symmetry, ⊿( L ) = In L /π 2 + E if 1 ≪ L ≪ L max (as in the gaussian orthogonal ensemble). When L ≫ L max , ⊿( L ) saturates non-universally at a value, determined by short classical orbits, of order ℏ –(N–1) for integrable systems and In (ℏ -1 ) for chaotic systems. These results are obtained by using the periodic-orbit expansion for the spectral density, together with classical sum rules for the intensities of long orbits and a semiclassical sum rule restricting the manner in which their contributions interfere. For two examples ⊿(L) is studied in detail: the rectangular billiard (integrable), and the Riemann zeta function (assuming its zeros to be the eigenvalues of an unknown quantum system whose unknown classical limit is chaotic).

Semiclassical Theory of Spectral Rigidity

http://www.cs.cas.cz/mp/papers07/causality-IT-final.pdf

Causality detection based on information-theoretic approaches in time series analysis

Ergodic theory concerns with the study of the long-time behavior of a dynamical system. An interesting result known as Birkhoff’s ergodic theorem states that under certain conditions, the time average exists and is equal to the space average. The applications of ergodic theory are the main concern of this note. We will introduce fundamental concepts in ergodic theory, Birkhoff’s ergodic theorem and its consequences.

An Introduction to Ergodic Theory

The persistent homology of a stationary point process on $\mathbf{R}^{N}$ is studied in this paper. As a generalization of continuum percolation theory, we study higher dimensional topological features of the point process such as loops, cavities, etc. in a multiscale way. The key ingredient is the persistence diagram, which is an expression of the persistent homology. We prove the strong law of large numbers for persistence diagrams as the window size tends to infinity and give a sufficient condition for the support of the limiting persistence diagram to coincide with the geometrically realizable region. We also discuss a central limit theorem for persistent Betti numbers.

/pdf/limit-theorems-for-persistence-diagrams-392riqv8u8.pdf

Limit theorems for persistence diagrams

We study the limiting behavior of Gaussian beta ensembles in the regime where $$\beta n = const$$
 as $$n \rightarrow \infty $$
 . The results are (1) Gaussian fluctuations for linear statistics of the eigenvalues, and (2) Poisson convergence of the bulk statistics. (2) is an alternative proof of the result by Benaych-Georges and Peche (J Stat Phys 161(3):633–656, 2015) with the explicit form of the intensity measure.

Gaussian Beta Ensembles at High Temperature: Eigenvalue Fluctuations and Bulk Statistics

We establish the strong law of large numbers for Betti numbers of random Cech complexes built on $${\mathbb {R}}^N$$
 -valued binomial point processes and related Poisson point processes in the thermodynamic regime. Here we consider both the case where the underlying distribution of the point processes is absolutely continuous with respect to the Lebesgue measure on $${\mathbb {R}}^N$$
 and the case where it is supported on a $$C^1$$
 compact manifold of dimension strictly less than N. The strong law is proved under very mild assumption which only requires that the common probability density function belongs to $$L^p$$
 spaces, for all $$1\le p < \infty $$
 .

/pdf/strong-law-of-large-numbers-for-betti-numbers-in-the-27yj2xfscb.pdf

Strong Law of Large Numbers for Betti Numbers in the Thermodynamic Regime

The paper describes the global limiting behavior of Gaussian beta ensembles where the parameter $$\beta $$
 is allowed to vary with the matrix size n. In particular, we show that as $$n \rightarrow \infty $$
 with $$n\beta \rightarrow \infty $$
 , the empirical distribution converges weakly to the semicircle distribution, almost surely. The Gaussian fluctuation around the limit is then derived by a martingale approach.

Khanh Duy Trinh

Papers

An Introduction to Ergodic Theory

Limit theorems for persistence diagrams

Gaussian Beta Ensembles at High Temperature: Eigenvalue Fluctuations and Bulk Statistics

Strong Law of Large Numbers for Betti Numbers in the Thermodynamic Regime

Global Spectrum Fluctuations for Gaussian Beta Ensembles: A Martingale Approach