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24 pages, 1413 KiB

Open AccessArticle

Extreme Value Analysis for Mixture Models with Heavy-Tailed Impurity

by Ekaterina Morozova and Vladimir Panov

Mathematics 2021, 9(18), 2208; https://doi.org/10.3390/math9182208 - 9 Sep 2021

Cited by 4 | Viewed by 1841

This paper deals with the extreme value analysis for the triangular arrays which appear when some parameters of the mixture model vary as the number of observations grows. When the mixing parameter is small, it is natural to associate one of the components [...] Read more.

This paper deals with the extreme value analysis for the triangular arrays which appear when some parameters of the mixture model vary as the number of observations grows. When the mixing parameter is small, it is natural to associate one of the components with “an impurity” (in the case of regularly varying distribution, “heavy-tailed impurity”), which “pollutes” another component. We show that the set of possible limit distributions is much more diverse than in the classical Fisher–Tippett–Gnedenko theorem, and provide the numerical examples showing the efficiency of the proposed model for studying the maximal values of the stock returns. Full article

(This article belongs to the Special Issue Advances in Applications of Probability Theory and Stochastic Processes)

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11 pages, 298 KiB

Open AccessFeature PaperArticle

On the Notion of Reproducibility and Its Full Implementation to Natural Exponential Families

by Shaul K. Bar-Lev

Mathematics 2021, 9(13), 1568; https://doi.org/10.3390/math9131568 - 3 Jul 2021

Viewed by 1668

Abstract

Let

F = \{F_{θ} : θ \in Θ \subset R\}

be a family of probability distributions indexed by a parameter

θ

and let

X_{1}, \dots, X_{n}

be i.i.d. r.v.’s with

L (X_{1}) =

[...] Read more.

Let

F = \{F_{θ} : θ \in Θ \subset R\}

be a family of probability distributions indexed by a parameter

θ

and let

X_{1}, \dots, X_{n}

be i.i.d. r.v.’s with

L (X_{1}) =

F_{θ} \in F

. Then,

F

is said to be reproducible if for all

θ \in Θ

and

n \in N

, there exists a sequence

{(α_{n})}_{n \geq 1}

and a mapping

g_{n} : Θ \to Θ,

θ ⟼ g_{n} (θ)

such that

L (α_{n} \sum_{i = 1}^{n} X_{i}) = F_{g_{n} (θ)} \in F .

In this paper, we prove that a natural exponential family

F

is reproducible iff it possesses a variance function which is a power function of its mean. Such a result generalizes that of Bar-Lev and Enis (1986, The Annals of Statistics) who proved a similar but partial statement under the assumption that

F

is steep as and under rather restricted constraints on the forms of

α_{n}

and

g_{n} (θ)

. We show that such restrictions are not required. In addition, we examine various aspects of reproducibility, both theoretically and practically, and discuss the relationship between reproducibility, convolution and infinite divisibility. We suggest new avenues for characterizing other classes of families of distributions with respect to their reproducibility and convolution properties . Full article

(This article belongs to the Special Issue Advances in Applications of Probability Theory and Stochastic Processes)

3 pages, 205 KiB

Open AccessArticle

A Variant of the Necessary Condition for the Absolute Continuity of Symmetric Multivariate Mixture

by Evgeniy Anatolievich Savinov

Mathematics 2021, 9(13), 1505; https://doi.org/10.3390/math9131505 - 27 Jun 2021

Viewed by 1215

Abstract

Sufficient conditions are given under which the absolute continuity of the joint distribution of conditionally independent random variables can be violated. It is shown that in the case of a dimension

n > 1

this occurs for a sufficiently large number of discontinuity [...] Read more.

Sufficient conditions are given under which the absolute continuity of the joint distribution of conditionally independent random variables can be violated. It is shown that in the case of a dimension

n > 1

this occurs for a sufficiently large number of discontinuity points of one-dimensional conditional distributions. Full article

(This article belongs to the Special Issue Advances in Applications of Probability Theory and Stochastic Processes)

15 pages, 564 KiB

Open AccessArticle

A High Order Accurate and Effective Scheme for Solving Markovian Switching Stochastic Models

by Yang Li, Taitao Feng, Yaolei Wang and Yifei Xin

Mathematics 2021, 9(6), 588; https://doi.org/10.3390/math9060588 - 10 Mar 2021

Cited by 1 | Viewed by 1813

Abstract

In this paper, we propose a new weak order 2.0 numerical scheme for solving stochastic differential equations with Markovian switching (SDEwMS). Using the Malliavin stochastic analysis, we theoretically prove that the new scheme has local weak order 3.0 convergence rate. Combining the special [...] Read more.

In this paper, we propose a new weak order 2.0 numerical scheme for solving stochastic differential equations with Markovian switching (SDEwMS). Using the Malliavin stochastic analysis, we theoretically prove that the new scheme has local weak order 3.0 convergence rate. Combining the special property of Markov chain, we study the effects from the changes of state space on the convergence rate of the new scheme. Two numerical experiments are given to verify the theoretical results. Full article

(This article belongs to the Special Issue Advances in Applications of Probability Theory and Stochastic Processes)

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13 pages, 365 KiB

Open AccessArticle

Cox Processes Associated with Spatial Copula Observed through Stratified Sampling

by Walguen Oscar and Jean Vaillant

Mathematics 2021, 9(5), 524; https://doi.org/10.3390/math9050524 - 3 Mar 2021

Cited by 1 | Viewed by 1728

Abstract

Cox processes, also called doubly stochastic Poisson processes, are used for describing phenomena for which overdispersion exists, as well as Poisson properties conditional on environmental effects. In this paper, we consider situations where spatial count data are not available for the whole study [...] Read more.

Cox processes, also called doubly stochastic Poisson processes, are used for describing phenomena for which overdispersion exists, as well as Poisson properties conditional on environmental effects. In this paper, we consider situations where spatial count data are not available for the whole study area but only for sampling units within identified strata. Moreover, we introduce a model of spatial dependency for environmental effects based on a Gaussian copula and gamma-distributed margins. The strength of dependency between spatial effects is related with the distance between stratum centers. Sampling properties are presented taking into account the spatial random field of covariates. Likelihood and Bayesian inference approaches are proposed to estimate the effect parameters and the covariate link function parameters. These techniques are illustrated using Black Leaf Streak Disease (BLSD) data collected in Martinique island. Full article

(This article belongs to the Special Issue Advances in Applications of Probability Theory and Stochastic Processes)

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17 pages, 679 KiB

Open AccessArticle

Towards Tensor Representation of Controlled Coupled Markov Chains

by Daniel McInnes, Boris Miller, Gregory Miller and Sergei Schreider

Mathematics 2020, 8(10), 1712; https://doi.org/10.3390/math8101712 - 5 Oct 2020

Cited by 1 | Viewed by 2006

Abstract

For a controlled system of coupled Markov chains, which share common control parameters, a tensor description is proposed. A control optimality condition in the form of a dynamic programming equation is derived in tensor form. This condition can be reduced to a system [...] Read more.

For a controlled system of coupled Markov chains, which share common control parameters, a tensor description is proposed. A control optimality condition in the form of a dynamic programming equation is derived in tensor form. This condition can be reduced to a system of coupled ordinary differential equations and admits an effective numerical solution. As an application example, the problem of the optimal control for a system of water reservoirs with phase and balance constraints is considered. Full article

(This article belongs to the Special Issue Advances in Applications of Probability Theory and Stochastic Processes)

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41 pages, 501 KiB

Open AccessArticle

The Feynman–Kac Representation and Dobrushin–Lanford–Ruelle States of a Quantum Bose-Gas

by Yuri Suhov, Mark Kelbert and Izabella Stuhl

Mathematics 2020, 8(10), 1683; https://doi.org/10.3390/math8101683 - 1 Oct 2020

Viewed by 1757

Abstract

This paper focuses on infinite-volume bosonic states for a quantum particle system (a quantum gas) in

R^{d}

. The kinetic energy part of the Hamiltonian is the standard Laplacian (with a boundary condition at the border of a ‘box’). The particles interact [...] Read more.

This paper focuses on infinite-volume bosonic states for a quantum particle system (a quantum gas) in

R^{d}

. The kinetic energy part of the Hamiltonian is the standard Laplacian (with a boundary condition at the border of a ‘box’). The particles interact with each other through a two-body finite-range potential depending on the distance between them and featuring a hard core of diameter

a > 0

. We introduce a class of so-called FK-DLR functionals containing all limiting Gibbs states of the system. As a justification of this concept, we prove that for

d = 2

, any FK-DLR functional is shift-invariant, regardless of whether it is unique or not. This yields a quantum analog of results previously achieved by Richthammer. Full article

(This article belongs to the Special Issue Advances in Applications of Probability Theory and Stochastic Processes)

23 pages, 348 KiB

Open AccessArticle

Rate of Convergence and Periodicity of the Expected Population Structure of Markov Systems that Live in a General State Space

by P. -C. G. Vassiliou

Mathematics 2020, 8(6), 1021; https://doi.org/10.3390/math8061021 - 22 Jun 2020

Cited by 5 | Viewed by 2090

Abstract

In this article we study the asymptotic behaviour of the expected population structure of a Markov system that lives in a general state space (MSGS) and its rate of convergence. We continue with the study of the asymptotic periodicity of the expected population [...] Read more.

In this article we study the asymptotic behaviour of the expected population structure of a Markov system that lives in a general state space (MSGS) and its rate of convergence. We continue with the study of the asymptotic periodicity of the expected population structure. We conclude with the study of total variability from the invariant measure in the periodic case for the expected population structure of an MSGS. Full article

(This article belongs to the Special Issue Advances in Applications of Probability Theory and Stochastic Processes)

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