Absolute continuity of the images of measures

One presents a new method for isolating conditions for the absolute continuity of distributions of functionals of random processes. The obtained general result is applied to the investigation of integral functionals of stationary processes, to the supremum of semistable processes, and to the norm of stable Banach-valued vectors.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 142, pp. 48–54, 1985.     

Absolute continuity of vector-valued self-affine measures

This paper is devoted to the absolute continuity of (scalar-valued or vector-valued) self-affine measures and their properties on the boundary of an invariant set. We first extend the definition of WSC to self-affine IFS, and then obtain a necessary and sufficient condition for the vector-valued self-affine measures to be absolutely continuous with respect to the Lebesgue measure. In addition, we prove that, for any IFS and any invariant open set V, the corresponding (scalar-valued or vector-valued) invariant measure is supported either in V or in ∂V.     

Absolute continuity of some semi-selfdecomposable distributions and self-similar measures

P.S. numbers are introduced in relation to absolute continuity of some infinite Bernoulli convolutions. Absolute continuity and continuous singularity of some semi-selfdecomposable distributions are studied as marginal distributions of subordinators. It is shown that these properties are widely different according as their spans are P.V. numbers or the reciprocals of P.S. numbers. A simple example of a subordinator whose distribution is continuous singular for small time and absolutely continuous for large time is given. Absolute continuity of convolutions of multidimensional homogeneous self-similar measures are also discussed. Received: 27 December 1998 / Revised version: 20 August 1999 / Published online: 31 May 2000     

Absolute continuity,singularity, and supports of Gauss semigroups on a Lie group

A Gauss semigroupS on a connected Lie group is absolutely continuous if and only if a certain differential operator associated withS is hypoelliptic. OtherwiseS is singular. IfS is absolutely continuous it has remarkable differentiability properties. Moreover the supports of the measures inS are described. The general results are specialized to the group of affine mappings on the real line.     

Estimation of Palm measures of stationary point processes

Summary Estimators of the Palm measure of a stationary point process on a finite-dimensional Euclidean space are developed and shown to be strongly uniformly consistent. From them, similarly consistent estimators of reduced moment measures, the spectral measure, the spectral density function and the underlying probability measure itself are derived. Normal and Poisson approximations to distributions of estimators are presented. Application is made to the problem of combined inference and linear state estimation.Research supported by Air Force Office of Scientific Research, AFSC, grant 82-0029C. The United States Government is authorized to reproduce and distribute reprints for governmental purposes     

Absolute continuity and local limit theorems for homogeneous functionals of point processes

Lithuanian Mathematical Journal - We study the absolute continuity and local limit theorems for homogeneous functionals defined on configurations of point processes (p.p.s). For empirical p.p.s, we...     

Absolute continuity of convolutions of orbital measures on Riemannian symmetric spaces

We study the absolute continuity of the measures and of on the Riemannian symmetric spaces X of noncompact type for nonzero elements Xj, X∈a. For m,l?r+1, where r is the rank of X, the considered convolutions have a density. We conjecture that the condition m,l?r+1 is necessary. The conjecture is proved for the symmetric spaces of type An−1. Moreover, the minimal value of l is determined, in function of the irregularity of X.     

Absolute continuity of measures in the class of Markov and semi-Markov processes of diffusion type

The property of absolute continuity of measures in the class of semi-Markov processes of diffusion type is investigated. The measure of such a process can be represented in the form of a composition of two measures. The first one is the distribution of a random track, and the second one is a conditional distribution of the time run along the track. The desired density (if it exists) is represented in the form of the product of the corresponding two densities. The first density is based on the asymptotic of the distribution density of the first exit point for the process exiting from an ellipsoidal neighborhood of its initial point. In terms of the associated Markov process and the induced Wiener process, this formula coincides with the known formula for the density of a diffusion-type Markov process measure. The second density is based on the semi-Markov property, which implies that the conditional distribution of the time run along a given track is the distribution of a monotone process with independent increments. Bibliography: 6 titles.Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 294, 2002, pp. 216–244.This research was supported by the Russian Foundation for Basic Research, grant 01-01-00613, and by the Program Leading Scientific Schools, grant 00-15-96019.Translated by B. P. Harlamov.     

On a singularity occurring in a self-correcting point process model

In a self-correcting point process model a boundary point of the parameter set is shown to be singular. This means a local behavior of the model which is qualitatively different from the LAN (or LAMN) condition satisfied at the other parameter points. As a consequence we obtain a nonnormal limiting distribution of the ML-estimator normalized with the random Fisher information.Work supported by a Heisenberg grant of the Deutsche Forschungsgemeinschaft.     

Absolute continuity of information channels

Let [X, v, Y] be an abstract information channel with the input X = (X, ) and the output Y = (Y, ) which are measurable spaces, and denote by L(Y) = L(Y, ) the Banach space of all bounded signed measures with finite total variation as norm. The channel distribution ν(·,·) is considered as a function defined on (X, ) and valued in L(Y). It will be proved that, if the measurable space (Y, ) is countably generated, then the is a strongly measurable function from X into L(Y) if and only if there exists a probability measure μ on (Y, ) which dominates every measure ν(x, ·) (x X). Furthermore, under this condition, the Radon-Nikodym derivative ν(x, dy)/μ(dy) is jointly measurable with respect to the product measure space (X, , m) (Y, , μ) where m is any but fixed probability measure of (X, ). As an application, it will be shown that the channel given as above is uniformly approximated by channels of Hibert-Schmidt type.     

Absolute continuity of spectral shift

In this paper we develop the method of double operator integrals to prove trace formulae for functions of contractions, dissipative operators, unitary operators and self-adjoint operators. To establish the absolute continuity of spectral shift, we use the Sz.-Nagy theorem on the absolute continuity of the spectral measure of the minimal unitary dilation of a completely nonunitary contraction. We also give a construction of an intermediate contraction for a pair of contractions with trace class difference.     

Absolute continuity of Wiener processes

By a Wiener process we mean a countably additive random measure taking independent values on disjoint sets. Given two continuous Wiener processes we give their decomposition into weakly equivalent and mutually singular parts.     

Hyperbolic stochastic differential equations: Absolute continuity of the LKW of the solution at a fixed point

LetW be the Wiener process onT=[0, 1]². Consider the stochastic integral equation $$\begin{gathered} X_\zeta = x_0 + \int_{R_\zeta } {a_1 (\zeta \prime )X(s\prime ,dt\prime )ds\prime + } \int_{R_\zeta } {a_2 (\zeta \prime )X(ds\prime ,t\prime )dt\prime } \hfill \\ + \int_{R_\zeta } {a_3 (X_{\zeta \prime , } \zeta \prime )W(ds\prime ,dt\prime ) + } \int_{R_\zeta } {a_4 (X_{\zeta \prime , } \zeta \prime )ds\prime ,dt\prime ,} \hfill \\ \end{gathered} $$ whereR _ζ =(s, t) ∈ T, andx ₀ ∈ ?. Under some assumptions on the coefficients a_i, the existence and uniqueness of a solution for this stochastic integral equation is already known (see [6]). In this paper we present some sufficient conditions for the law ofX _ζ to have a density.     

Absolute continuity of a vector-valued measure

The necessary and sufficient condition is proved for absolute continuity of a vector-valued measure in the sense of Bartle.     

向量值函数的绝对连续性

本文将强绝对连续性和绝对连续性两个概念推广到取值于任意拓扑向量空间的函数,和将弱绝对连续性推广到取值于局部凸空间的函数.描述了这些概念之间的关系及特征,并推广了马绍群的一些结果.     

Absolute continuity of operator-self-decomposable distributions on Rd

It is shown that every genuinely d-dimensional operator-self-decomposable distribution is absolutely continuous.