Weighted isoperimetric and Poincaré-type inequalities are studied for κ-concave probability measures (in the hierarchy of convex measures).

Let K be an origin symmetric star body in ℝⁿ. We prove, under very mild conditions on the function f : [0, ∞)→ℝ, that if the function f(‖x‖_K) is positive definite on ℝⁿ, then the space (ℝⁿ, ‖ ⋅ ‖_K) embeds isometrically in L₀. This generalizes the solution to Schoenberg’s problem and leads to progress in characterization of n-dimensional versions, i.e. random vectors X=(X₁, …, X_n) in ℝⁿ such that the random variables ∑a_iX_i are identically distributed for all a∈ℝⁿ, up to a constant depending on ‖a‖_K only.

We study how well moments of sums of independent symmetric random variables with logarithmically concave tails may be approximated by moments of Gaussian random variables.

We provide general formulas to compute the expectations of absolute value and sign of Gaussian quadratic forms, i.e. $\mathbb{E}$ |〈X, AX〉+〈b, X〉+c| and $\mathbb{E}$ sgn(〈X, AX〉+〈b, X〉+c) for centered Gaussian random vector X, fixed matrix A, vector b and constant c. Products of Gaussian quadratics are also discussed and followed with several interesting applications.

We consider a parametric class T_γ of expanding maps of [0, 1] with a neutral fixed point at 0 for which there exists an unique invariant absolutely continuous probability measure ν_γ on [0, 1]. On the probability space ([0, 1], ν_γ), we prove the weak invariance principle for the partial sums of f○T_γⁱ in some special cases involving non-standard normalization. We also prove new moment inequalities and exponential bounds for the partial sums of f○T_γⁱ when f is some Hölder function such that f(0)=ν_γ(f).

Necessary and sufficient conditions for the classical central limit theorem (CLT) for i.i.d. random vectors in an arbitrary separable Banach space require not only assumptions on the original distribution, but also on the sample. What we do here is to continue our study of the CLT in terms of the original distribution. Of course, some new ingredient must be introduced, so we allow slight modifications of the random vectors. In particular, we restrict our modifications to be continuous, and to be no larger than a fixed small number, or in some cases a fixed small proportion of the magnitude of the individual elements of the sample. We find that if we use certain interpolation space norms to measure the magnitude of such modifications, then the CLT can be improved. Examples of our result are also included.

We study weak convergence of general (smoothed) empirical processes indexed by classes of functions $\mathcal{F}$ under minimal conditions. We present a general result that, applied to specific situations, enables us to prove uniform central limit theorems under P-pregaussian assumption on $\mathcal{F}$ only.

We provide bounds for the VC dimension of class of sets formed by unions, intersections, and products of VC classes of sets $\mathcal{C}$₁, …, $\mathcal{C}$_m.

The limit behavior is studied for the distributions of normalized U- and V-statistics of an arbitrary order with canonical (degenerate) kernels, based on samples of increasing sizes from a stationary sequence of observations satisfying φ- or α-mixing. The case of m-dependent sequences is separately studied. The corresponding limit distributions are represented as infinite multilinear forms of a centered Gaussian sequence with a known covariance matrix. Moreover, under φ-mixing, exponential inequalities are obtained for the distribution tails of these statistics with bounded kernels.

This paper is concerned with the study of functional limit theorems constructed from multiple generations of a supercritical branching process. The results we present include infinite dimensional functional laws of Strassen and Chung-Wichura type in the space (C₀[0, 1])^∞.

The purpose of this paper is to synthesize the approaches taken by Chatterjee-Meckes and Reinert-Röllin in adapting Stein’s method of exchangeable pairs for multivariate normal approximation. The more general linear regression condition of Reinert-Röllin allows for wider applicability of the method, while the method of bounding the solution of the Stein equation due to Chatterjee-Meckes allows for improved convergence rates. Two abstract normal approximation theorems are proved, one for use when the underlying symmetries of the random variables are discrete, and one for use in contexts in which continuous symmetry groups are present. A first application is presented to projections of exchangeable random vectors in ℝⁿ onto one-dimensional subspaces. The application to runs on the line from Reinert-Röllin is reworked to demonstrate the improvement in convergence rates, and a new application to joint value distributions of eigenfunctions of the Laplace-Beltrami operator on a compact Riemannian manifold is presented.

We point out that the method of Davis-Mikosch [Ann. Probab. 27 (1999) 522–536] gives for a symmetric circulant n×n matrix composed of i.i.d. entries with mean 0 and finite (2+δ)-moments in the first half-row that the maximum eigenvalue is on the order $\sqrt{2n \log n}$, and the fluctuations are Gumbel.

Let X₁, X₂, …, X_n, … be a sequence of iid random variables with values in a finite ordered alphabet {α₁, …, α_m}. Let LI_n be the length of the longest increasing subsequence of X₁, X₂, …, X_n. Properly centered and normalized, the limiting distribution of LI_n is expressed as various functionals of m and (m−1)-dimensional Brownian motions. These expressions are then related to similar functionals appearing in queueing theory, allowing us to further describe asymptotic behaviors when, in turn, m grows without bound. The finite alphabet results are then used to treat the countable (infinite) alphabet case.

This paper considers random (non-Hermitian) circulant matrices, and proves several results analogous to recent theorems on non-Hermitian random matrices with independent entries. In particular, the limiting spectral distribution of a random circulant matrix is shown to be complex normal, and bounds are given for the probability that a circulant sign matrix is singular.

Conditional expectations of a fractional Brownian motion with Hurst index H respect to the filtration of a fractional Brownian motion with Hurst index H^′, both contained in the fractional Brownian field, are studied. A stochastic integral representation of those processes is constructed from the covariance structure of the underlying fractional Brownian field. As processes, the conditional expectations contain martingale components and for dual pairs of Hurst indices the processes become pure martingales which, up to a multiplicative constant, coincide with the fundamental martingales of fractional Brownian motions.

We characterize stochastic compactness and convergence in distribution of a Lévy process at “large times", i.e., as t→∞, by properties of its associated Lévy measure, using a mechanism for transferring between discrete (random walk) and continuous time results. We thereby obtain also domain of attraction characterisations for the process at large times. As an illustration of the stochastic compactness ideas, semi-stable laws are considered.

Let G={G(x), x∈R₊}, G(0)=0, be a mean zero Gaussian process with E(G(x)−G(y))²=σ²(x−y). Let ρ(x)=½ d²/dx² σ²(x), x≠0. When ρ^k is integrable at zero and satisfies some additional regularity conditions,

lim_h↓0∫ : ((G(x+h)−G(x))/h)^k : g(x) dx= : (G')^k : (g) a.s.

for all g∈$\mathcal{B}$₀(R⁺), the set of bounded Lebesgue measurable functions on R₊ with compact support. Here G' is a generalized derivative of G and : ( ⋅ )^k : is the k–th order Wick power.

In this paper we obtain a Bernstein type inequality for a class of weakly dependent and bounded random variables. The proofs lead to a moderate deviations principle for sums of bounded random variables with exponential decay of the strong mixing coefficients that complements the large deviation result obtained by Bryc and Dembo (1998) under superexponential mixing rates.

We obtain the limit distribution of the test statistic of the most powerful invariant test for location families of densities. As an application, we obtain the consistency of this test. From these results similar results are obtained for the test statistic of the most powerful invariant test for scale families.

We consider pointwise consistency properties of kernel regression function type estimators where the bandwidth sequence is not necessarily deterministic. In some recent papers uniform convergence rates over compact sets have been derived for such estimators via empirical process theory. We now show that it is possible to get optimal results in the pointwise case as well. The main new tool for the present work is a general moment bound for empirical processes which may be of independent interest.

The problem of estimation of integral curves of a vector field based on its noisy observations is studied. For Nadaraya-Watson type estimators, several results on asymptotics of the shortest distance from the estimated curve to a specified region have been proved. The problem is motivated by applications in diffusion tensor imaging where it is of importance to test various hypotheses of geometric nature based on the estimated distances.

Given an i.i.d. sample from the law ℙ on the unit circle, we obtain uniform central limit theorems for the random measures induced by trigonometric series and sieved maximum likelihood density estimators. The limit theorems are uniform over balls in Sobolev-Hilbert spaces of order s>1/2.