Perturbation analysis of functionals of random measures

1995 ◽  
Vol 27 (2) ◽  
pp. 306-325 ◽  
Author(s):  
François Baccelli ◽  
Maurice Klein ◽  
Sergei Zuyev
Keyword(s):  
Phase Space ◽  
Random Measure ◽  
Random Processes ◽  
Stationary Processes ◽  
Light Traffic ◽  
Random Measures ◽  
First Order ◽  
Order Expansion ◽  
Palm Measure ◽  
New Algorithms

We use the fact that the Palm measure of a stationary random measure is invariant to phase space change to generalize the light traffic formula initially obtained for stationary processes on a line to general spaces. This formula gives a first-order expansion for the expectation of a functional of the random measure when its intensity vanishes. This generalization leads to new algorithms for estimating gradients of functionals of geometrical random processes.

Perturbation analysis of functionals of random measures

1995 ◽  
Vol 27 (02) ◽  
pp. 306-325 ◽  
Author(s):  
François Baccelli ◽  
Maurice Klein ◽  
Sergei Zuyev
Keyword(s):  
Phase Space ◽  
Random Measure ◽  
Random Processes ◽  
Stationary Processes ◽  
Light Traffic ◽  
Random Measures ◽  
First Order ◽  
Order Expansion ◽  
Palm Measure ◽  
New Algorithms

We use the fact that the Palm measure of a stationary random measure is invariant to phase space change to generalize the light traffic formula initially obtained for stationary processes on a line to general spaces. This formula gives a first-order expansion for the expectation of a functional of the random measure when its intensity vanishes. This generalization leads to new algorithms for estimating gradients of functionals of geometrical random processes.

On Product Measures Associated with Stationary Processes

2018 ◽  
Author(s):  
Alain Boudou ◽  
Yves Romain
Keyword(s):  
Type Theorem ◽  
Random Measure ◽  
Continuous Homomorphism ◽  
Stationary Processes ◽  
Convolution Product ◽  
Random Measures ◽  
Unknown Quantity ◽  
Shift Operators ◽  
Potential Applications ◽  
Product Measures

This article considers the connections between product measures and stationary processes. It first provides an overview of historical facts and relevant terminology, basic concepts and the mathematical approach. In particular, it discusses random measures, the projection-valued spectral measure (PVSM), convolution products, and the association between shift operators and PVSMs. It then presents the main results and their first potential applications, focusing on stochastic integrals, the image of a random measure under measurable mapping, the existence of a transport-type theorem, and the transpose of a continuous homomorphism between groups. It also describes the PVSM associated with a unitary operator, the convolution product of two PVSMs, the unitary operators generated by a PVSM, extension of the convolution product of two PVSMs, an equation where the unknown quantity is a PVSM, and the convolution product of two random measures. The article concludes with an analysis of mathematical developments related to the previous results.

A remark on the construction of random measures

1979 ◽  
Vol 85 (1) ◽  
pp. 111-115 ◽  
Author(s):  
J. Mecke
Keyword(s):  
Phase Space ◽  
Point Process ◽  
Point Processes ◽  
Polish Space ◽  
Random Measure ◽  
Process Theory ◽  
Sufficient Condition ◽  
Conditional Probabilities ◽  
Random Measures ◽  
Finite Dimensional

In his paper (2), Ripley has shown that some of the basic results of point process theory can be established without making any use of topological assumptions concerning the phase space. For the construction of distributions of point processes he needs pseudo-topological assumptions. In the present paper, another sufficient condition is given for the fact that a distribution of a random measure can be constructed by help of finite-dimensional distributions. This condition is expressed by purely measure-theoretic notions. Roughly speaking, it is supposed that concerning the existence of conditional probabilities the phase space behaves as if it were a Polish space. Measureable spaces with such a property are called ‘full measurable spaces’; they are identical with the measurable spaces of type (B) in (1).

scholarly journals On intensities of modulated Cox measures

1998 ◽  
Vol 11 (3) ◽  
pp. 411-423 ◽  
Author(s):  
Jewgeni H. Dshalalow
Keyword(s):  
Stochastic Process ◽  
Stochastic Models ◽  
Queueing Systems ◽  
Random Measure ◽  
Random Measures ◽  
Explicit Formulas ◽  
State Dependent

In this paper we introduce and study functionals of the intensities of random measures modulated by a stochastic process ξ, which occur in applications to stochastic models and telecommunications. Modulation of a random measure by ξ is specified for marked Cox measures. Particular cases of modulation by ξ as semi-Markov and semiregenerative processes enabled us to obtain explicit formulas for the named intensities. Examples in queueing (systems with state dependent parameters, Little's and Campbell's formulas) demonstrate the use of the results.

scholarly journals Light-cone distribution amplitudes of light mesons with QED effects

2021 ◽  
Vol 2021 (11) ◽  
Author(s):  
Martin Beneke ◽  
Philipp Böer ◽  
Jan-Niklas Toelstede ◽  
K. Keri Vos
Keyword(s):  
Light Cone ◽  
Electromagnetic Coupling ◽  
First Order ◽  
Order Expansion ◽  
Group Evolution ◽  
Fixed Order ◽  
Light Mesons ◽  
Charged Mesons ◽  
Qed Effects ◽  
Exclusive Processes

Abstract We discuss the generalization of the leading-twist light-cone distribution amplitude for light mesons including QED effects. This generalization was introduced to describe virtual collinear photon exchanges at the strong-interaction scale ΛQCD in the factorization of QED effects in non-leptonic B-meson decays. In this paper we study the renormalization group evolution of this non-perturbative function. For charged mesons, in particular, this exhibits qualitative differences with respect to the well-known scale evolution in QCD only, especially regarding the endpoint-behaviour. We analytically solve the evolution equation to first order in the electromagnetic coupling αem, which resums large logarithms in QCD on top of a fixed-order expansion in αem. We further provide numerical estimates for QED corrections to Gegenbauer coefficients as well as inverse moments relevant to (QED-generalized) factorization theorems for hard exclusive processes.

scholarly journals A VARIATIONAL FORMULATION OF SYMPLECTIC NONCOMMUTATIVE MECHANICS

2007 ◽  
Vol 04 (05) ◽  
pp. 789-805 ◽  
Author(s):  
IGNACIO CORTESE ◽  
J. ANTONIO GARCÍA
Keyword(s):  
Dynamical System ◽  
Phase Space ◽  
Variational Principle ◽  
Configuration Space ◽  
Variational Formulation ◽  
Noncommutative Space ◽  
First Order ◽  
Dynamical Properties ◽  
Definition Of ◽  
Space Noncommutativity

The standard lore in noncommutative physics is the use of first order variational description of a dynamical system to probe the space noncommutativity and its consequences in the dynamics in phase space. As the ultimate goal is to understand the inherent space noncommutativity, we propose a variational principle for noncommutative dynamical systems in configuration space, based on results of our previous work [18]. We hope that this variational formulation in configuration space can be of help to elucidate the definition of some global and dynamical properties of classical and quantum noncommutative space.

Rarefactions of compound point processes

1984 ◽  
Vol 21 (04) ◽  
pp. 710-719
Author(s):  
Richard F. Serfozo
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Classical Result ◽  
Point Processes ◽  
Rare Events ◽  
Random Measure ◽  
Bernoulli Trials ◽  
Random Measures ◽  
Infinitely Divisible ◽  
Independent Increments

The Poisson process is regarded as a point process of rare events because of the classical result that the number of successes in a sequence of Bernoulli trials is asymptotically Poisson as the probability of a success tends to 0. It is shown that this rareness property of the Poisson process is characteristic of any infinitely divisible point process or random measure with independent increments. These processes and measures arise as limits of certain rarefactions of compound point processes: purely atomic random measures with uniformly null atom sizes. Examples include thinnings and partitions of point processes.

Limit theorems for maxima and crossings of a sequence of Gaussian processes and approximation of random processes

1991 ◽  
Vol 28 (01) ◽  
pp. 17-32 ◽  
Author(s):  
O. V. Seleznjev
Keyword(s):  
Gaussian Processes ◽  
Limit Distribution ◽  
Limit Theorems ◽  
Point Processes ◽  
Random Processes ◽  
Trigonometric Polynomials ◽  
Stationary Processes ◽  
Periodic Processes

We consider the limit distribution of maxima and point processes, connected with crossings of an increasing level, for a sequence of Gaussian stationary processes. As an application we investigate the limit distribution of the error of approximation of Gaussian stationary periodic processes by random trigonometric polynomials in the uniform metric.

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