A central limit theorem for measurements on the logarithmic scale and its application to dimension estimates

We show consistency and asymptotic normality of certain estimators for expected exponential growth rates under i.i.d. observations. These statistical functionals are of the form

T(F)=∫log∫h(x,y)F(dx)F(dy)     

Sets versus trial sequences,Hausdorff versus von Mises: “Pure” mathematics prevails in the foundations of probability around 1920

The paper discusses the tension which occurred between the notions of set (with measure) and (trial-) sequence (or—to a certain degree—between nondenumerable and denumerable sets) when used in the foundations of probability theory around 1920. The main mathematical point was the logical need for measures in order to describe general nondiscrete distributions, which had been tentatively introduced before (1919) based on von Mises’s notion of the “Kollektiv.” In the background there was a tension between the standpoints of pure mathematics and “real world probability” (in the words of J.L. Doob) at the time. The discussion and publication in English translation (in Appendix) of two critical letters of November 1919 by the “pure” mathematician Felix Hausdorff to the engineer and applied mathematician Richard von Mises compose about one third of the paper. The article also investigates von Mises’s ill-conceived effort to adopt measures and his misinterpretation of an influential book of Constantin Carathéodory. A short and sketchy look at the subsequent development of the standpoints of the pure and the applied mathematician—here represented by Hausdorff and von Mises—in the probability theory of the 1920s and 1930s concludes the paper.     

The central limit theorem for Riesz-Raikov sums

Summary Letf be a real-valued function with period 1 satisfying some regularity conditions. It will be proved that, for any ₁,..., _d > 1, the distribution of on the probability space ([0, 1],dt) converges to a normal distribution whose covariance is given by algebraic relations among the _i 's. This generalizes the classical work by M. Kac and refines the characterization off to have a degenerate limit. It also shows that the limit law of is in most cases a Cauchy distribution.     

An intermediate Baum-Katz theorem

We extend the classical Hsu-Robbins-Erd?s theorem to the case when all moments exist, but the moment generating function does not, viz., we assume that Eexp{(log⁺|X|)^α}<∞ for some α>1. We also present multi-index versions of the same and of a related result due to Lanzinger in which the assumption is that Eexp{|X|^α}<∞ for some α∈(0,1).     

Local limit theorems for multiplicative free convolutions

This paper describes the quality of convergence to an infinitely divisible law relative to free multiplicative convolution. We show that convergence in distribution for products of identically distributed and infinitesimal free random variables implies superconvergence of their probability densities to the density of the limit law. Superconvergence to the marginal law of free multiplicative Brownian motion at a specified time is also studied. In the unitary case, the superconvergence to free Brownian motion and that to the Haar measure are shown to be uniform over the entire unit circle, implying further a free entropic limit theorem and a universality result for unitary free Lévy processes. Finally, the method of proofs on the positive half-line gives rise to a new multiplicative Boolean to free Bercovici–Pata bijection.     

An infinite dimensional convolution theorem with applications to the efficient estimation of the integrated volatility

This paper proposes a general approach to obtain asymptotic lower bounds for the estimation of random functionals. The main result is an abstract convolution theorem in a non parametric setting, based on an associated LAMN property. This result is then applied to the estimation of the integrated volatility, or related quantities, of a diffusion process, when the diffusion coefficient depends on an independent Brownian motion.     

The emergence of open sets,closed sets,and limit points in analysis and topology

General topology has its roots in real and complex analysis, which made important uses of the interrelated concepts of open set, of closed set, and of a limit point of a set. This article examines how those three concepts emerged and evolved during the late 19th and early 20th centuries, thanks especially to Weierstrass, Cantor, and Lebesgue. Particular attention is paid to the different forms of the Bolzano–Weierstrass Theorem found in the latter's unpublished lectures. An abortive early, unpublished introduction of open sets by Dedekind is examined, as well as how Peano and Jordan almost introduced that concept. At the same time we study the interplay of those three concepts (together with those of the closure of a set and of the derived set of a set) in the struggle to determine the ultimate foundations on which general topology was built, during the first half of the 20th century.     

Chainability and Hemmingsen's theorem

On the surface, the definitions of chainability and Lebesgue covering dimension ?1 are quite similar as covering properties. Using the ultracoproduct construction for compact Hausdorff spaces, we explore the assertion that the similarity is only skin deep. In the case of dimension, there is a theorem of E. Hemmingsen that gives us a first-order lattice-theoretic characterization. We show that no such characterization is possible for chainability, by proving that if κ is any infinite cardinal and A is a lattice base for a nondegenerate continuum, then A is elementarily equivalent to a lattice base for a continuum Y, of weight κ, such that Y has a 3-set open cover admitting no chain open refinement.     

Rates of convergence in the central limit theorem for linear statistics of martingale differences

In this paper, we give rates of convergence for minimal distances between linear statistics of martingale differences and the limiting Gaussian distribution. In particular the results apply to the partial sums of (possibly long range dependent) linear processes, and to the least squares estimator in some parametric regression models.     

A central limit theorem for stationary random fields

Summary. We prove a central limit theorem for strictly stationary random fields under a projective assumption. Our criterion is similar to projective criteria for stationary sequences derived from Gordin's theorem about approximating martingales. However our approach is completely different, for we establish our result by adapting Lindeberg's method. The criterion that it provides is weaker than martingale-type conditions, and moreover we obtain as a straightforward consequence, central limit theorems for α-mixing or φ-mixing random fields. Received: 19 February 1997 / In revised form: 2 September 1997     

Law of large numbers and central limit theorem for randomly forced PDE's

We consider a class of dissipative PDE's perturbed by an external random force. Under the condition that the distribution of perturbation is sufficiently non-degenerate, a strong law of large numbers (SLLN) and a central limit theorem (CLT) for solutions are established and the corresponding rates of convergence are estimated. It is also shown that the estimates obtained are close to being optimal. The proofs are based on the property of exponential mixing for the problem in question and some abstract SLLN and CLT for mixing-type Markov processes.     

Rate of escape and central limit theorem for the supercritical Lamperti problem

The study of discrete-time stochastic processes on the half-line with mean drift at x$x$ given by _μ1(x)→0${\mu }_1\left(x\right)\to 0$ as x→∞$x\to \infty$ is known as Lamperti’s problem  . We give sharp almost-sure bounds for processes of this type in the case where _μ1(x)${\mu }_1\left(x\right)$ is of order x^−β$x^{-\beta }$ for some β∈(0,1)$\beta \in (0,1)$. The bounds are of order t^1/(1+β)$t^{1/(1+\beta )}$, so the process is super-diffusive but sub-ballistic (has zero speed). We make minimal assumptions on the moments of the increments of the process (finiteness of (2+2β+ε)$(2+2\beta +\epsilon )$-moments for our main results, so fourth moments certainly suffice) and do not assume that the process is time-homogeneous or Markovian. In the case where x^β_μ1(x)$x^{\beta }{\mu }_1\left(x\right)$ has a finite positive limit, our results imply a strong law of large numbers, which strengthens and generalizes earlier results of Lamperti and Voit. We prove an accompanying central limit theorem, which appears to be new even in the case of a nearest-neighbour random walk, although our result is considerably more general. This answers a question of Lamperti. We also prove transience of the process under weaker conditions than those that we have previously seen in the literature. Most of our results also cover the case where β=0$\beta =0$. We illustrate our results with applications to birth-and-death chains and to multi-dimensional non-homogeneous random walks.     

The central limit theorem for random perturbations of rotations

Summary.   We prove a functional central limit theorem for stationary random sequences given by the transformations

A central limit theorem for martingales and an application to branching processes

A functional central limit theorem is obtained for martingales which are not uniformly asymptotically negligible but grow at a geometric rate. The function space is not the usual C[0,1] or D[0,1] but R^N, the space of all real sequences and the metric used leads to a non-separable metric space.The main theorem is applied to a martingale obtained from a supercritical Galton-Watson branching process and as simple corollaries the already known central limit theorems for the Harris and Lotka-Nagaev estimators of the mean of the offspring distribution, are obtained.     

A retelling of Newton’s work on Kepler’s Laws

This paper details an investigation into Kepler’s Laws. Newton’s technique for deducing an inverse-square law from Kepler’s Laws is given a modern presentation, with necessary background material included. Kepler’s Laws are then deduced from the assumption of an inverse-square law. This is done in a geometric style, inspired by Newton’s work. Finally, a problem involving planetary orbits is stated and solved using the earlier results of the paper.     

Henriksen and Isbell on f-rings

This paper gives an account of the contributions of Melvin Henriksen and John Isbell to the abstract theory of f-rings and formally real f-rings, with particular attention to the manner in which their work was framed by universal algebra. I describe the origins of the Pierce-Birkhoff Conjecture and present some other unsolved problems suggested by their work.