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A Riemann approach to random variation

Patrick Muldowney (2006)

Mathematica Bohemica

This essay outlines a generalized Riemann approach to the analysis of random variation and illustrates it by a construction of Brownian motion in a new and simple manner.

A second order SDE for the Langevin process reflected at a completely inelastic boundary

Jean Bertoin (2008)

Journal of the European Mathematical Society

It was shown in [2] that a Langevin process can be reflected at an energy absorbing boundary. Here, we establish that the law of this reflecting process can be characterized as the unique weak solution to a certain second order stochastic differential equation with constraints, which is in sharp contrast with a deterministic analog.

A sharp analysis on the asymptotic behavior of the Durbin–Watson statistic for the first-order autoregressive process

Bernard Bercu, Frédéric Proïa (2013)

ESAIM: Probability and Statistics

The purpose of this paper is to provide a sharp analysis on the asymptotic behavior of the Durbin–Watson statistic. We focus our attention on the first-order autoregressive process where the driven noise is also given by a first-order autoregressive process. We establish the almost sure convergence and the asymptotic normality for both the least squares estimator of the unknown parameter of the autoregressive process as well as for the serial correlation estimator associated with the driven noise....

A sharp maximal inequality for continuous martingales and their differential subordinates

Adam Osękowski (2013)

Czechoslovak Mathematical Journal

Assume that X , Y are continuous-path martingales taking values in ℝ ν , ν ≥ 1 , such that Y is differentially subordinate to X . The paper contains the proof of the maximal inequality ∥ sup t ≥ 0 | Y t | ∥ 1 ≤ 2 ∥ sup t ≥ 0 | X t | ∥ 1 . The constant 2 is shown to be the best possible, even in the one-dimensional setting of stochastic integrals with respect to a standard Brownian motion. The proof uses Burkholder’s method and rests on the construction of an appropriate special function.

A simple approach to functional inequalities for non-local Dirichlet forms

Jian Wang (2014)

ESAIM: Probability and Statistics

With direct and simple proofs, we establish Poincaré type inequalities (including Poincaré inequalities, weak Poincaré inequalities and super Poincaré inequalities), entropy inequalities and Beckner-type inequalities for non-local Dirichlet forms. The proofs are efficient for non-local Dirichlet forms with general jump kernel, and also work for Lp(p> 1) settings. Our results yield a new sufficient condition for fractional Poincaré inequalities, which were recently studied in [P.T. Gressman,...

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