A theorem of McCann [15] shows that for any two absolutely continuous probability measures on Rd there exists a monotone transformation sending one probability measure to the other. A consequence of this theorem, relevant to statistics, is that density estimation can be recast in terms of transformations. In particular, one can fix any absolutely continuous probability measure, call it P, and t...

We give a new proof of a version of Klein's theorem on the existence of absolutely continuous spectrum for the Anderson model on the Bethe Lattice at weak disorder. It is widely believed that the Anderson model [An] should exhibit absolutely continuous spectrum at weak disorder in dimensions three and higher. But it is only for the Bethe lattice B, or Cayley tree, that this has been established...

Abstract. We prove several new results on the absolutely continuous spectra of perturbed one-dimensional Stark operators. First, we find new classes of perturbations, characterized mainly by smoothness conditions, which preserve purely absolutely continuous spectrum. Then we establish stability of the absolutely continuous spectrum in more general situations, where imbedded singular spectrum ma...

to demonstrate more visibly the close relation between thecontinuity and integrability, a new proof for the banach-zareckitheorem is presented on the basis of the radon-nikodym theoremwhich emphasizes on measure-type properties of the lebesgueintegral. the banach-zarecki theorem says that a real-valuedfunction $f$ is absolutely continuous on a finite closed intervalif and only if it is continuo...

In mathematical statistics courses, students learn that the quadratic function is minimized when x mean of random variable X, and graphs this for any two distributions X are simply translates each other. We focus on problem minimizing defined by in context mixtures probability discrete, absolutely continuous, singular continuous types. This important, example, Bayesian statistics, one attempts ...

The states ρi, i = 1, 2 in the state space S of a C-algebra A are absolutely continuous if and only if there exist absolutely continuous probability measures μi on S such that ρi is the barycenter of μi. This technique allows one to study the transformation of conditional expectations under an absolutely continuous change of state using the classical Bayes formula, which can be exploited to obt...

We investigate the existence of an absolutely continuous martingale measure. For continuous processes we show that the absence of arbitrage for general admissible integrands implies the existence of an absolutely continuous (not necessarily equivalent) local martingale measure. We also rephrase Radon-Nikodym theorems for predictable processes. 1.Introduction. In our paper Delbaen and Schacherma...