/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

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

Convergence of Probability Measures

Introduction to linear and nonlinear programming

(1987). Applied Probability and Queues. Journal of the Operational Research Society: Vol. 38, No. 11, pp. 1095-1096.

https://link.springer.com/content/pdf/10.1057%2Fjors.1987.184.pdf

Applied Probability and Queues

We study a certain random groeth model in two dimensions closely related to the one-dimensional totally asymmetric exclusion process. The results show that the shape fluctuations, appropriately scaled, converges in distribution to the Tracy-Widom largest eigenvalue distribution for the Gaussian Unitary Ensemble.

Shape Fluctuations and Random Matrices

We develop Stein's method for exchangeable pairs for a rich class of distributional approximations including the Gaussian distributions as well as the non-Gaussian limit distributions. As a consequence we obtain convergence rates in limit theorems of partial sums for certain sequences of dependent, identically distributed random variables which arise naturally in statistical mechanics, in particular in the context of the Curie-Weiss models. Our results include a Berry-Esseen rate in the Central Limit Theorem for the total magnetization in the classical Curie-Weiss model, for high temperatures as well as at the critical temperature, where the Central Limit Theorem fails. Moreover, we analyze Berry-Esseen bounds as the temperature converges to one and obtain a threshold for the speed of this convergence. Single spin distributions satisfying the Griffiths-Hurst-Sherman (GHS) inequality like models of liquid helium or continuous Curie-Weiss models are considered.

/pdf/stein-s-method-for-dependent-random-variables-occuring-in-3j1tegxtwx.pdf

Stein's Method for Dependent Random Variables Occuring in Statistical Mechanics

The purpose of the present paper is to establish moderate deviation principles for a rather general class of random variables fulfilling certain bounds of the cumulants. We apply a celebrated lemma of the theory of large deviations probabilities due to Rudzkis, Saulis, and Statulevicius. The examples of random objects we treat include dependency graphs, subgraph-counting statistics in Erdos–Renyi random graphs and U-statistics. Moreover, we prove moderate deviation principles for certain statistics appearing in random matrix theory, namely characteristic polynomials of random unitary matrices and the number of particles in a growing box of random determinantal point processes such as the number of eigenvalues in the GUE or the number of points in Airy, Bessel, and sine random point fields.

Moderate Deviations via Cumulants

We combine Malliavin calculus with Stein's method to derive bounds for the Variance-Gamma approximation of functionals of isonormal Gaussian processes, in particular of random variables living inside a fixed Wiener chaos induced by such a process. The bounds are presented in terms of Malliavin operators and norms of contractions. We show that a sequence of distributions of random variables in the second Wiener chaos converges to a Variance-Gamma distribution if and only if their moments of order two to six converge to that of a Variance-Gamma distributed random variable (six moment theorem). Moreover, simplified versions for Laplace or symmetrized Gamma distributions are presented. Also multivariate extensions and a universality result for homogeneous sums are considered.

/pdf/malliavin-stein-method-for-variance-gamma-approximation-on-38xayeh3m2.pdf

Malliavin-Stein method for variance-gamma approximation on Wiener space

We construct the conditional version of $k$ independent and identically distributed random walks on $R$ given that they stay in strict order at all times. This is a generalisation of so-called non-colliding or non-intersecting random walks, the discrete variant of Dyson's Brownian motions, which have been considered yet only for nearest-neighbor walks on the lattice. Our only assumptions are moment conditions on the steps and the validity of the local central limit theorem. The conditional process is constructed as a Doob $h$-transform with some positive regular function $V$ that is strongly related with the Vandermonde determinant and reduces to that function for simple random walk. Furthermore, we prove an invariance principle, i.e., a functional limit theorem towards Dyson's Brownian motions, the continuous analogue.

Ordered Random Walks

We derive necessary and sufficient conditions for a sum of i.i.d. random variables – where , but – to satisfy a moderate deviations principle. Moreover we show that this equivalence is a typical moderate deviations phenomenon. It is not true in a large deviations regime.

/pdf/moderate-deviations-for-i-i-d-random-variables-8ovlfyj5ta.pdf

Peter Eichelsbacher

Papers

Stein's Method for Dependent Random Variables Occuring in Statistical Mechanics

Moderate Deviations via Cumulants

Malliavin-Stein method for variance-gamma approximation on Wiener space

Ordered Random Walks

Moderate deviations for i.i.d. random variables