Given a stationary process (Xp)p∈Z and an event B ∈ σ(Xp, p ∈ Z), we study the almost sure convergence as n and m go to infinity of the “bilateral” martingale E [1B |X−n, X−n+1, . . . , Xm−1, Xm ] . We show that almost sure convergence holds in some classical examples such as i.i.d. or Markov processes, as well as for the natural generator of Chacon’s transformation. However, we also prove that...

Concentration inequalities have been the subject of exciting developments during the last two decades, and have been intensively studied and used as a powerful tool in various areas. These include convex geometry, functional analysis, statistical physics, mathematical statistics, pure and applied probability theory (e.g., concentration of measure phenomena in random graphs, random matrices, and...

We prove new inequalities implying exponential decay of relative entropy functionals for a class of Zero–Range processes on the complete graph. We first consider the case of uniformly increasing rates, where we use a discrete version of the Bakry– Emery criterium to prove spectral gap and entropy dissipation estimates, uniformly over the number of particles and the number of vertices. We then s...

We extend the results of Walters on the uniqueness of invariant measures with maximal entropy on compact groups to an arbitrary locally compact group. We show that the maximal entropy is attained at the left Haar measure and the measure of maximal entropy is unique.

Abstract It is generally understood that a given one-dimensional diffusion may be transformed by Cameron–Martin–Girsanov measure change into another with the same volatility but different drift. But to achieve this, we have know change-of-measure local martingale write down true martingale. We provide complete characterisation of when this happens. This enables us discuss absence arbitrage in g...

Contents 1. Introduction 1 2. Ergodic theory: some background 4 3. Entropy of dynamical systems: some more background 6 4. Conditional Expectation and Martingale theorems 12 5. Countably generated σ-algebras and Conditional measures 14 6. Leaf-wise Measures, the construction 19 7. Leaf-wise Measures and entropy 37 8. The product structure 61 9. Invariant measures and entropy for higher rank sub...

This paper reevaluates the mathematical and economic meaning of no arbitrage in frictionless markets. Contrary to the traditional view, no arbitrage is not generally equivalent to the existence of an equivalent martingale measure. Departures from this equivalence allow asset prices to contain a monetary component. The refined view is that no arbitrage and no private monetary value components ar...