THE SMOOTHNESS OF ORBITAL MEASURES ON NONCOMPACT SYMMETRIC SPACES

Let $G/K$ be an irreducible symmetric space, where G is a noncompact, connected Lie group and K is a compact, connected subgroup. We use decay properties of the spherical functions to show that the convolution product of any $r=r(G/K)$ continuous orbital measures has its density function in $L^{2}(G)$ and hence is an absolutely continuous measure with respect to the Haar measure. The number r is approximately the rank of $G/K$ . For the special case of the orbital measures, $\nu _{a_{i}}$ , supported on the double cosets $Ka_{i}K$ , where $a_{i}$ belongs to the dense set of regular elements, we prove the sharp result that $\nu _{a_{1}}\ast \nu _{a_{2}}\in L^{2},$ except for the symmetric space of Cartan class $AI$ when the convolution of three orbital measures is needed (even though $\nu _{a_{1}}\ast \nu _{a_{2}}$ is absolutely continuous).

Asymptotics for the Nodal Components of Non-Identically Distributed Monochromatic Random Waves

We study monochromatic random waves on ${\mathbb{R}}^n$ defined by Gaussian variables whose variances tend to zero sufficiently fast. This has the effect that the Fourier transform of the monochromatic wave is an absolutely continuous measure on the sphere with a suitably smooth density, which connects the problem with the scattering regime of monochromatic waves. In this setting, we compute the asymptotic distribution of the nodal components of random monochromatic waves, showing that the number of nodal components contained in a large ball $B_R$ grows asymptotically like $R/\pi $ with probability $p_n>0$ and is bounded uniformly in $R$ with probability $1-p_n$ (which is positive if and only if $n\geqslant 3$). In the latter case, we show the existence of a unique noncompact nodal component. We also provide an explicit sufficient stability criterion to ascertain when a more general Gaussian probability distribution has the same asymptotic nodal distribution law.

Semi-discrete optimal transport: a solution procedure for the unsquared Euclidean distance case

We consider the problem of finding an optimal transport plan between an absolutely continuous measure and a finitely supported measure of the same total mass when the transport cost is the unsquared Euclidean distance. We may think of this problem as closest distance allocation of some resource continuously distributed over Euclidean space to a finite number of processing sites with capacity constraints. This article gives a detailed discussion of the problem, including a comparison with the much better studied case of squared Euclidean cost. We present an algorithm for computing the optimal transport plan, which is similar to the approach for the squared Euclidean cost by Aurenhammer et al. (Algorithmica 20(1):61–76, 1998) and Mérigot (Comput Graph Forum 30(5):1583–1592, 2011). We show the necessary results to make the approach work for the Euclidean cost, evaluate its performance on a set of test cases, and give a number of applications. The later include goodness-of-fit partitions, a novel visual tool for assessing whether a finite sample is consistent with a posited probability density.

The Mathematics of the Martingale Approach

In this chapter we present the two main mathematical results which are needed for the application of the martingale approach to pricing and hedging. We first discuss and prove the martingale representation theorem which says that in a Wiener framework, every martingale can be represented as a stochastic integral. We then discuss and prove the Girsanov Theorem which gives us control over the class of absolutely continuous measure transformations. The abstract theory is then applied to stochastic differential equations, and to maximum likelihood estimation.

Characterizing the Absolute Continuity of the Convolution of Orbital Measures in a Classical Lie Algebra

Let 𝓰 be a compact simple Lie algebra of dimension d. It is a classical result that the convolution of any d non-trivial, G-invariant, orbitalmeasures is absolutely continuous with respect to Lebesgue measure on 𝓰, and the sum of any d non-trivial orbits has non-empty interior. The number d was later reduced to the rank of the Lie algebra (or rank +1 in the case of type An). More recently, the minimal integer k = k(X) such that the k-fold convolution of the orbital measure supported on the orbit generated by X is an absolutely continuous measure was calculated for each X ∈ 𝓰.In this paper 𝓰 is any of the classical, compact, simple Lie algebras. We characterize the tuples (X1 , . . . , XL), with Xi ∊ 𝓰, which have the property that the convolution of the L-orbital measures supported on the orbits generated by the Xi is absolutely continuous, and, equivalently, the sum of their orbits has non-empty interior. The characterization depends on the Lie type of 𝓰 and the structure of the annihilating roots of the Xi. Such a characterization was previously known only for type An.

W-LIKE MAPS WITH VARIOUS INSTABILITIES OF ACIM'S

This paper generalizes the results of [Li et al., 2011] and then provides an interesting example. We construct a family of W-like maps {Wa} with a turning fixed point having slope s1 on one side and –s2 on the other. Each Wa has an absolutely continuous invariant measure μa. Depending on whether [Formula: see text] is larger, equal or smaller than 1, we show that the limit of μa is a singular measure, a combination of singular and absolutely continuous measure or an absolutely continuous measure, respectively. It is known that the invariant density of a single piecewise expanding map has a positive lower bound on its support. In Sec. 4 we give an example showing that in general, for a family of piecewise expanding maps with slopes larger than 2 in modulus and converging to a piecewise expanding map, their invariant densities do not necessarily have a uniform positive lower bound on the support.

Sequential selection of random vectors under a sum constraint

We observe a sequence X1, X2,…, Xn of independent and identically distributed coordinatewise nonnegative d-dimensional random vectors. When a vector is observed it can either be selected or rejected but once made this decision is final. In each coordinate the sum of the selected vectors must not exceed a given constant. The problem is to find a selection policy that maximizes the expected number of selected vectors. For a general absolutely continuous distribution of the Xi we determine the maximal expected number of selected vectors asymptotically and give a selection policy which asymptotically achieves optimality. This problem raises a question closely related to the following problem. Given an absolutely continuous measure μ on Q = [0,1]d and a τ ∈ Q, find a set A of maximal measure μ(A) among all A ⊂ Q whose center of gravity lies below τ in all coordinates. We will show that a simplicial section {x ∈ Q | 〈x, θ〉 ≤ 1}, where θ ∈ ℝd, θ ≥ 0, satisfies a certain additional property, is a solution to this problem.

Sequential selection of random vectors under a sum constraint

We observe a sequence X 1, X 2,…, X n of independent and identically distributed coordinatewise nonnegative d-dimensional random vectors. When a vector is observed it can either be selected or rejected but once made this decision is final. In each coordinate the sum of the selected vectors must not exceed a given constant. The problem is to find a selection policy that maximizes the expected number of selected vectors. For a general absolutely continuous distribution of the X i we determine the maximal expected number of selected vectors asymptotically and give a selection policy which asymptotically achieves optimality. This problem raises a question closely related to the following problem. Given an absolutely continuous measure μ on Q = [0,1] d and a τ ∈ Q, find a set A of maximal measure μ(A) among all A ⊂ Q whose center of gravity lies below τ in all coordinates. We will show that a simplicial section { x ∈ Q | 〈 x , θ 〉 ≤ 1}, where θ ∈ ℝ d , θ ≥ 0, satisfies a certain additional property, is a solution to this problem.