A Note on Wiener–Hopf Factorization for Markov Additive Processes

We prove the Wiener–Hopf factorization for Markov additive processes. We derive also Spitzer–Rogozin theorem for this class of processes which serves for obtaining Kendall’s formula and Fristedt representation of the cumulant matrix of the ladder epoch process. Finally, we also obtain the so-called ballot theorem.     

Introduction—nonlinear dynamics in urban and regional modeling

The Laguerre transform, introduced by Keilson and Nunn (1979) and Keilson, Nunn, Sumita (1981), provides an algorithmic basis for the computation of multiple convolutions in conjunction with other algebraic and summation operations. The methods enable one to evaluate numerically a variety of results in applied probability and statistics that have been available only formally. For certain more complicated models, the formulation must be extended. In this paper we establish the matrix Laguerre transform, appropriate for the study of semi-Markov processes and Markov renewal processes, as an extension of the scalar Laguerre transform. The new formalism enables one to calculate matrix convolutions and other algebraic operations in matrix form. As an application, a matrix renewal function is evaluated and its limit theorem is numerically exhibited.     

一类马尔可夫骨架过程的强大数定律和中心极限定理

基于马尔可夫骨架过程极限分布的已有研究结果,本文运用波莱尔-康特立引理、更新理论、科尔莫哥洛夫的强大数定律以及独立同分布情形的中心极限定理等重要理论,分别给出了一类马尔可夫骨架过程对应的累积过程满足强大数定律和中心极限定理的充分条件.     

马尔可夫骨架过程的极限分布

本文运用基本更新定理和Smith关键更新定理等理论和方法,对马尔可夫骨架过程的极限分布进行深入研究,得到主要结果如下:去掉了原有结果中要求的绝对连续的条件,给出了马尔可夫骨架过程极限分布存在的充分条件;得到了马尔可夫骨架过程极限分布的具体公式,并证明了该极限分布为概率分布.     

关于树指标非齐次马氏链的广义熵遍历定理

主要研究了树指标非齐次马氏链的广义熵遍历定理.首先证明了树指标非齐次马氏链上的二元函数延迟平均的强极限定理.然后得到了树指标非齐次马氏链上状态出现延迟频率的强大数定律,以及树指标非齐次马氏链的广义熵遍历定理.作为推论,推广了一些已有结果.同时,证明了局部有限无穷树树指标有限状态随机过程广义熵密度的一致可积性.     

The Geometric Markov Renewal Processes with Application to Finance

We introduce the geometric Markov renewal processes as a model for a security market and study this processes in a series scheme. We consider its approximations in the form of averaged, merged and double averaged geometric Markov renewal processes. Weak convergence analysis and rates of convergence of ergodic geometric Markov renewal processes are presented. Martingale properties, infinitesimal operators of geometric Markov renewal processes are presented and a Markov renewal equation for expectation is derived. As an application, we consider the case of two ergodic classes. Moreover, we consider a generalized binomial model for a security market induced by a position dependent random map as a special case of a geometric Markov renewal process.     

Strong Law of Large Numbers and Central Limit Theorems for Functionals of Inhomogeneous Semi-Markov Processes

Limit theorems for functionals of classical (homogeneous) Markov renewal and semi-Markov processes have been known for a long time, since the pioneering work of Pyke Schaufele (Limit theorems for Markov renewal processes, Ann. Math. Statist., 35(4):1746–1764, 1964). Since then, these processes, as well as their time-inhomogeneous generalizations, have found many applications, for example, in finance and insurance. Unfortunately, no limit theorems have been obtained for functionals of inhomogeneous Markov renewal and semi-Markov processes as of today, to the best of the authors’ knowledge. In this article, we provide strong law of large numbers and central limit theorem results for such processes. In particular, we make an important connection of our results with the theory of ergodicity of inhomogeneous Markov chains. Finally, we provide an application to risk processes used in insurance by considering a inhomogeneous semi-Markov version of the well-known continuous-time Markov chain model, widely used in the literature.     

A strong log-concavity property for measures on Boolean algebras

We introduce the antipodal pairs property for probability measures on finite Boolean algebras and prove that conditional versions imply strong forms of log-concavity. We give several applications of this fact, including improvements of some results of Wagner, a new proof of a theorem of Liggett stating that ultra-log-concavity of sequences is preserved by convolutions, and some progress on a well-known log-concavity conjecture of J. Mason.     

A Capelli type theorem for multiplicative convolutions of polynomials

We prove a Capelli type theorem on the canonical decomposition for multiplicative convolutions of polynomials. We derive then some irreducibility criteria for convolutions of polynomials in several variables over a given field. The irreducibility conditions are expressed only in terms of the degrees of the polynomials in convolution, the degrees of their coefficients, and the degrees of some suitable divisors of the resulting leading coefficient. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Applications of P-adic generalized functions and approximations by a system of p-adic translations of a function

Under some conditions we prove that the convergence of a sequence of functions in the space of P-adic generalized functions is equivalent to its convergence in the space of locally integrable functions. Some analogs are established of the Wiener tauberian theorem and the Wiener theorem on denseness of translations for P-adic convolutions and translations.     

Bessel Convolutions on Matrix Cones: Algebraic Properties and Random Walks

Bessel-type convolution algebras of measures on the matrix cones of positive semidefinite q×q-matrices over ?,?,? were introduced recently by Rösler. These convolutions depend on a continuous parameter, generate commutative hypergroups, and have Bessel functions of matrix argument as characters. In this paper, we study the algebraic structure of these hypergroups. In particular, the subhypergroups, quotients, and automorphisms are classified. The algebraic properties are partially related to the properties of random walks on these matrix Bessel hypergroups. In particular, known properties of Wishart distributions, which form Gaussian convolution semigroups on these hypergroups, are put into a new light. Moreover, limit theorems for random walks are presented. In particular, we obtain strong laws of large numbers and a central limit theorem with Wishart distributions as limits.     

Generalized convolutions and the Levi-Civita functional equation

In Borowiecka et al. (Bernoulli 21(4):2513–2551, 2015) the authors show that every generalized convolution can be used to define a Markov process, which can be treated as a Lévy process in the sense of this convolution. The Bessel process is the best known example here. In this paper we present new classes of regular generalized convolutions enlarging the class of such Markov processes. We give here a full characterization of such generalized convolutions \(\diamond \) for which \(\delta _x \diamond \delta _1\), \(x \in [0,1]\), is a convex linear combination of \(n=3\) fixed measures and only the coefficients of the linear combination depend on x. For \(n=2\) it was shown in Jasiulis-Goldyn and Misiewicz (J Theor Probab 24(3):746–755, 2011) that such a convolution is unique (up to the scale and power parameters). We show also that characterizing such convolutions for \(n \geqslant 3\) is equivalent to solving the Levi-Civita functional equation in the class of continuous generalized characteristic functions.     

A limit theorem for truncated measures in generalized convolution algebras

Carrying over a result of Kuelbs and Ledoux, we show that in generalized convolution algebras as introduced by Urbanik, domains of attraction of stable measures go over, after suitable truncation and renormalization, into domains of attraction of the characteristic measure of the convolution algebra. For convolutions which are induced by a (deterministic) semigroup operation, only the “large” terms are responsible for convergence to the stable law. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part III     

Liouville theorem and coupling on negatively curved Riemannian manifolds

By using probabilistic approaches, Liouville theorems are proved for a class of Riemannian manifolds with Ricci curvatures bounded below by a negative function. Indeed, for these manifolds we prove that all harmonic functions (maps) with certain growth are constant. In particular, the well-known Liouville theorem due to Cheng for sublinear harmonic functions (maps) is generalized. Moreover, our results imply the Brownian coupling property for a class of negatively curved Riemannian manifolds. This leads to a negative answer to a question of Kendall concerning the Brownian coupling property.     

The Key Renewal Theorem for a Transient Markov Chain

We consider a time-homogeneous real-valued Markov chain X _n , n≥0. Suppose that this chain is transient, that is, X _n generates a σ-finite renewal measure. We prove the key renewal theorem under the condition that this chain has jumps that are asymptotically homogeneous at infinity and asymptotically positive drift.     

On the Exceptional Set for Absolute Continuity of Bernoulli Convolutions

We prove that the set of exceptional \({\lambda\in (1/2,1)}\) such that the associated Bernoulli convolution is singular has zero Hausdorff dimension, and likewise for biased Bernoulli convolutions, with the exceptional set independent of the bias. This improves previous results by Erdös, Kahane, Solomyak, Peres and Schlag, and Hochman. A theorem of this kind is also obtained for convolutions of homogeneous self-similar measures. The proofs are very short, and rely on old and new results on the dimensions of self-similar measures and their convolutions, and the decay of their Fourier transform.     

Fourier transforms in function-space and parabolic differential equations

A Fourier transform for sufficiently bounded and regular functionals on C([0, 1], R) is introduced, which interchanges multiplication and differentiation. We define convolutions and prove a convolution theorem. Finally, we use the transform to uniquely solve the Cauchy problem for second-order parabolic equations in function-space, where the coefficients depend only on t.     

Algebras of Jordan brackets and generalized Poisson algebras

We construct bases for free unital generalized Poisson superalgebras and for free unital superalgebras of Jordan brackets. Also, we prove an analogue of Farkas’ theorem for PI unital generalized Poisson algebras and PI unital algebras of Jordan brackets. Relations between generic Poisson superalgebras and superalgebras of Jordan brackets are studied.     

On symmetric stochastic convolutions

The notion of stochastic convolution arises from a study on various operations on probability measures. Although these operations appear in diverse branches of mathematics, their properties turn out to be similar. A general model for the study of such convolutions has been provided in earlier papers by the author. In this paper we consider some new properties of this model for convolutions corresponding to generalized integral transforms. The tools of this model can be applied to study many realizations of stochastic convolutions, for example, Urbanik's regular convolutions, the polynomial hypergroup's convolutions of Lasser, the convolution of Askey-Gasper and Levitan's convolution.     

Verma modules over generalized Witt algebras

We constuct and investigate a structure of Verma-like modules over generalized Witt algebras. We also prove Futorny-like theorem for irreducible weight modlues whose dimensions of the weight spaces are uniformly bounded.