Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · math/0507526

Concentration inequalities with exchangeable pairs (Ph.D. thesis)

Sourav Chatterjee
View original source ↗Read source document ↗Export metadata ↓

Abstract

The purpose of this dissertation is to introduce a version of Stein's method of exchangeable pairs to solve problems in measure concentration. We specifically target systems of dependent random variables, since that is where the power of Stein's method is fully realized. Because the theory is quite abstract, we have tried to put in as many examples as possible. Some of the highlighted applications are as follows: (a) We shall find an easily verifiable condition under which a popular heuristic technique originating from physics, known as the "mean field equations" method, is valid. No such condition is currently known. (b) We shall present a way of using couplings to derive concentration inequalities. Although couplings are routinely used for proving decay of correlations, no method for using couplings to derive concentration bounds is available in the literature. This will be used to obtain (c) concentration inequalities with explicit constants under Dobrushin's condition of weak dependence. (d) We shall give a method for obtaining concentration of Haar measures using convergence rates of related random walks on groups. Using this technique and one of the numerous available results about rates of convergence of random walks, we will then prove (e) a quantitative version of Voiculescu's celebrated connection between random matrix theory and free probability.

Explore related subjects

math.PRmath-phmath.MP

Keep this discovery

BibTeXRIS

Sourav Chatterjee. 2005-07-26. Concentration inequalities with exchangeable pairs (Ph.D. thesis). https://arxiv.org/abs/math/0507526

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Averaging principles for nonautonomous multiscale stochastic Burgers equations with reflection

In this paper, we study averaging principles for nonautonomous multiscale stochastic Burgers equations with reflection. First, we derive a general averaging principle applicable to such equations under minimal assumptions. Subsequently, since the coefficients of the obtained averaged equation still depend on the small scaling parameter $\e$, we impose either periodic or asymptotic conditions on the coefficients, thereby obtain two distinct averaged equations whose coefficients are independent of $\e$ and establish two averaging principles. Stopping times and Khasminskii's time discretization schemes play an important role. Finally, a concrete example is provided to illustrate the applicability and validity of the theoretical results.

math.PR↗
arXiv2026

Spectral properties of Random Matrices

We give the theoretical foundations of random matrix theory through the definitions of a random matrix, a random probability measure and the corresponding empirical spectral distribution. The technical tool we use is the Stieltjes transform method through which we prove optimal convergence of the empirical spectral distribution of random sample covariance matrices to the deterministic Marchenko-Pastur distribution. We also give new results about the rigidity of the eigenvalues of this random sample covariance matrix and the rate of their convergence. We then define the Dyson equation method to prove new local laws about a random matrix model that interpolates between the Marchenko-Pastur distribution, the elliptical law and the circular law. Through our work these local laws can be considered universal.

math.PR↗
arXiv2026

Moments approach for the elephant random walk

We discuss the method of moments for the one-dimensional elephant random walk (ERW). We first derive a differential recurrence relation for the characteristic function of the ERW, which yields a corresponding system of recurrence relations for its moments. We then obtain asymptotic approximations for the moments in each of the three parameter regimes of the ERW. Finally, by establishing the convergence of the moments and verifying the corresponding moment-determinacy conditions, we identify the limiting distributions of the ERW in each regime.

math.PR↗