37 research outputs found
On the relationship of energy and probability in models of classical statistical physics
In this paper we present a new point of view on the mathematical foundations
of statistical physics of infinite volume systems. This viewpoint is based on
the newly introduced notions of transition energy function, transition energy
field and one-point transition energy field. The former of them, namely the
transition energy function, is a generalization of the notion of relative
Hamiltonian introduced by Pirogov and Sinai. However, unlike the (relative)
Hamiltonian, our objects are defined axiomatically by their natural and
physically well-founded intrinsic properties. The developed approach allowed us
to give a proper mathematical definition of the Hamiltonian without involving
the notion of potential, to propose a justification of the Gibbs formula for
infinite systems and to answer the problem stated by D. Ruelle of how wide the
class of specifications, which can be represented in Gibbsian form, is.
Furthermore, this approach establishes a straightforward relationship between
the probabilistic notion of (Gibbs) random field and the physical notion of
(transition) energy, and so opens the possibility to directly apply
probabilistic methods to the mathematical problems of statistical physics
On the relationship of energy and probability in models of classical statistical physics
In this paper we present a new point of view on the mathematical foundations of statistical physics of infinite volume systems. This viewpoint is based on the newly introduced notions of transition energy function, transition energy field and one-point transition energy field. The former of them, namely the transition energy function, is a generalization of the notion of relative Hamiltonian introduced by Pirogov and Sinai. However, unlike the (relative) Hamiltonian, our objects are defined axiomatically by their natural and physically well-founded intrinsic properties. The developed approach allowed us to give a proper mathematical definition of the Hamiltonian without involving the notion of potential, to propose a justification of the Gibbs formula for infinite systems and to answer the problem stated by D. Ruelle of how wide the class of specifications, which can be represented in Gibbsian form, is. Furthermore, this approach establishes a straightforward relationship between the probabilistic notion of (Gibbs) random field and the physical notion of (transition) energy, and so opens the possibility to directly apply probabilistic methods to the mathematical problems of statistical physics
On the relationship of energy and probability in models of classical statistical physics
In this paper we present a new point of view on the mathematical foundations of statistical physics of infinite volume systems. This viewpoint is based on the newly introduced notions of transition energy function, transition energy field and one-point transition energy field. The former of them, namely the transition energy function, is a generalization of the notion of relative Hamiltonian introduced by Pirogov and Sinai. However, unlike the (relative) Hamiltonian, our objects are defined axiomatically by their natural and physically well-founded intrinsic properties. The developed approach allowed us to give a proper mathematical definition of the Hamiltonian without involving the notion of potential, to propose a justification of the Gibbs formula for infinite systems and to answer the problem stated by D. Ruelle of how wide the class of specifications, which can be represented in Gibbsian form, is. Furthermore, this approach establishes a straightforward relationship between the probabilistic notion of (Gibbs) random field and the physical notion of (transition) energy, and so opens the possibility to directly apply probabilistic methods to the mathematical problems of statistical physics
Partially ordered models
We provide a formal definition and study the basic properties of partially
ordered chains (POC). These systems were proposed to model textures in image
processing and to represent independence relations between random variables in
statistics (in the later case they are known as Bayesian networks). Our chains
are a generalization of probabilistic cellular automata (PCA) and their theory
has features intermediate between that of discrete-time processes and the
theory of statistical mechanical lattice fields. Its proper definition is based
on the notion of partially ordered specification (POS), in close analogy to the
theory of Gibbs measure. This paper contains two types of results. First, we
present the basic elements of the general theory of POCs: basic geometrical
issues, definition in terms of conditional probability kernels, extremal
decomposition, extremality and triviality, reconstruction starting from
single-site kernels, relations between POM and Gibbs fields. Second, we prove
three uniqueness criteria that correspond to the criteria known as bounded
uniformity, Dobrushin and disagreement percolation in the theory of Gibbs
measures.Comment: 54 pages, 11 figures, 6 simulations. Submited to Journal of Stat.
Phy
Estimation of cusp location of stochastic processes: a survey
© 2018, Springer Science+Business Media B.V., part of Springer Nature. We present a review of some recent results on estimation of location parameter for several models of observations with cusp-type singularity at the change point. We suppose that the cusp-type models fit better to the real phenomena described usually by change point models. The list of models includes Gaussian, inhomogeneous Poisson, ergodic diffusion processes, time series and the classical case of i.i.d. observations. We describe the properties of the maximum likelihood and Bayes estimators under some asymptotic assumptions. The asymptotic efficiency of estimators are discussed as well and the results of some numerical simulations are presented. We provide some heuristic arguments which demonstrate the convergence of log-likelihood ratios in the models under consideration to the fractional Brownian motion
On Compound Poisson Processes Arising in Change-Point Type Statistical Models as Limiting Likelihood Ratios
Different change-point type models encountered in statistical inference for
stochastic processes give rise to different limiting likelihood ratio
processes. In a previous paper of one of the authors it was established that
one of these likelihood ratios, which is an exponential functional of a
two-sided Poisson process driven by some parameter, can be approximated (for
sufficiently small values of the parameter) by another one, which is an
exponential functional of a two-sided Brownian motion. In this paper we
consider yet another likelihood ratio, which is the exponent of a two-sided
compound Poisson process driven by some parameter. We establish, that similarly
to the Poisson type one, the compound Poisson type likelihood ratio can be
approximated by the Brownian type one for sufficiently small values of the
parameter. We equally discuss the asymptotics for large values of the parameter
and illustrate the results by numerical simulations
On the Goodness-of-Fit Tests for Some Continuous Time Processes
We present a review of several results concerning the construction of the
Cramer-von Mises and Kolmogorov-Smirnov type goodness-of-fit tests for
continuous time processes. As the models we take a stochastic differential
equation with small noise, ergodic diffusion process, Poisson process and
self-exciting point processes. For every model we propose the tests which
provide the asymptotic size and discuss the behaviour of the power
function under local alternatives. The results of numerical simulations of the
tests are presented.Comment: 22 pages, 2 figure
Оценивание и проверка гипотезы о малых изменениях интенсивности пуассоновского процесса
A model of Poissonian observations having a jump (changepoint
Оценивание и проверка гипотезы о малых изменениях интенсивности пуассоновского процесса
A model of Poissonian observations having a jump (changepoint