Consider a Markov chain $\{X_n\}_{n\ge 0}$ with an ergodic probability measure $π$. Let $Ψ$ a function on the state space of the chain, with $α$-tails with respect to $π$, $α\in (0,2)$. We find sufficient conditions on the probability transition to prove convergence in law of $N^{1/α}\sum_n^N Ψ(X_n)$ to a $α$-stable law. ``Martingale approximation'' approach and ``coupling'' approach give two different sets of conditions. We extend these results to continuous time Markov jump processes $X_t$, whose skeleton chain satisfies our assumptions. If waiting time between jumps has finite expectation, we prove convergence of $N^{-1/α}\int_0^{Nt} V(X_s) ds$ to a stable process. In the case of waiting times with infinite average, we prove convergence to a Mittag-Leffler process.
The recent study by B. De Bruyne, S. N. Majumdar, H. Orland and G. Schehr [arXiv:2110.07573], concerning the conditioning of the Brownian motion and of random walks on global dynamical constraints over a finite time-window $T$, is reformulated as a general framework for the 'microcanonical conditioning' of Markov processes on time-additive observables. This formalism is applied to various types of Markov processes, namely discrete-time Markov chains, continuous-time Markov jump processes and diffusion processes in arbitrary dimension. In each setting, the time-additive observable is also fully…
In this paper we study the almost sure central limit theorem started from a point for additive functionals of a stationary and ergodic Markov chain via a martingale approximation in the almost sure sense. As a consequence we derive the quenched CLT for additive functionals or reversible Markov chains.
We provide necessary and sufficient conditions for convergence of exponential integrals of Markov additive processes. Other than in the classical Lévy case studied by Erickson and Maller we have to distinguish between almost sure convergence and convergence in probability. Our proofs rely on recent results on perpetuities in a Markovian environment by Alsmeyer and Buckmann.
General Markov chains with a countably additive transition probability in arbitrary phase space are considered. Markov operators extend from the space of countably additive measures to the space of finitely additive measures. In the author's papers a theorem was earlier proved that if all invariant finitely additive measures of a Markov chain are countably additive, i.e. there are no invariant purely finitely additive measures, then their subspace is finite-dimensional and the Markov chain satisfies the Doob-Doeblin quasicompactness conditions. In the same paper, a partial inversion of this th…
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.
General Markov chains in an arbitrary phase space are considered in the framework of the operator treatment. Markov operators continue from the space of countably additive measures to the space of finitely additive measures. Cycles of measures generated by the corresponding operator are constructed, and algebraic operations on them are introduced. One of the main results obtained is that any cycle of finitely additive measures can be uniquely decomposed into the coordinate-wise sum of a cycle of countably additive measures and a cycle of purely finitely additive measures.We have proved theorem…
We consider general Markov chains with discrete time in an arbitrary measurable (phase) space and homogeneous in time. Markov chains are defined by the classical transition function which within the framework of the operator treatment generates a conjugate pair of linear Markov operators in the Banach space of measurable bounded functions and in the Banach space of bounded finite additive measures. It is proved that the well-known Doeblin condition $ (D) $ of ergodicity (quasi\-compactness) of the Markov chain is equivalent to the condition $ (*) $: all finitely additive invariant measures of …
In this paper we study the additive functionals of Markov chains via conditioning with respect to both past and future of the chain. We shall point out new sufficient projective conditions, which assure that the variance of partial sums of n consecutive random variables of a stationary Markov chain is linear in n. The paper also addresses the central limit theorem problem and is listing several open questions.
In this paper, we consider general Markov chains (MC), specified by the transition probability (kernel) $ P (x, E) $, finitely additive in the second argument. Such MC are studied within the framework of the functional operator treatment. The state space (phase space) of the MC $ X $ has any cardinality, and the sigma-algebra $ Σ$ is discrete, i.e. is the set of all subsets in $ X $. This construction of the phase space $ (X, Σ) $ allows us to decompose the Markov kernel $ P (x, E) $ into the sum of two components - countably additive and purely finitely additive in the second argument and mea…
Consider a one-sided Markov additive process with an upper and a lower barrier, where each can be either reflecting or terminating. For both defective and non-defective processes and all possible scenarios we identify the corresponding potential measures, which generalizes a number of results for one-sided Levy processes. The resulting rather neat formulas have various applications, and in particular they lead to quasi-stationary distributions of the corresponding processes.
In this paper we study the central limit theorem for additive functionals of stationary Markov chains with general state space by using a new idea involving conditioning with respect to both the past and future of the chain. Practically, we show that any additive functionals of a stationary and totally ergodic Markov chain with the variance of S_n divided by n uniformly bounded, satisfies a square root central limit theorem with a random centering. We do not assume that the Markov chain is irreducible and aperiodic. However, the random centering is not needed if the Markov chain satisfies stro…
In this paper, we study Markov chains (MC) on topological spaces within the framework of the operator approach. We extend the Markov operator from the space of countably additive measures to the space of finitely additive measures. Cesaro means for a Markov sequence of measures and their asymptotic behavior in the weak topology are considered. It is proved ergodic theorem that in order for the Cesaro means to converge weakly to some bounded regular finitely additive (or countably additive) measure it is necessary and sufficient that all invariant finitely additive measures are not separable fr…
We study properties of the Laplace transforms of non-negative additive functionals of Markov chains. We are namely interested in a multiplicative ergodicity property used in [18] to study bifurcating processes with ancestral dependence. We develop a general approach based on the use of the operator perturbation method. We apply our general results to two examples of Markov chains, including a linear autoregressive model. In these two examples the operator-type assumptions reduce to some expected finite moment conditions on the functional (no exponential moment conditions are assumed in this wo…
In this paper, we are concerned with centered Markov Additive Processes $\{(X_t,Y_t)\}_{t\in\T}$ where the driving Markov process $\{X_t\}_{t\in\T}$ has a finite state space. Under suitable conditions, we provide a local limit theorem for the density of the absolutely continuous part of the probability distribution of $t^{-1/2}Y_t$ given $X_0$. The rate of convergence and the moment condition are the expected ones with respect to the i.i.d case. An application to the joint distribution of local times of a finite jump process is sketched.
We study the exponential growth of bifurcating processes with ancestral dependence. We suppose here that the lifetimes of the cells are dependent random variables, that the numbers of new cells are random and dependent. Lifetimes and new cells's numbers are also assumed to be dependent. We illustrate our results by examples, including some Markov models. Our approach is related to the behaviour of the Laplace transform of nonnegative additive functional of Markov chains and require weak moment assumption (no exponential moment is needed).
Using the renewal approach we prove exponential inequalities for additive functionals and empirical processes of ergodic Markov chains, thus obtaining counterparts of inequalities for sums of independent random variables. The inequalities do not require functions of the chain to be bounded and moreover all the involved constants are given by explicit formulas whenever the usual drift condition holds, which may be of interest in practical applications e.g. to MCMC algorithms.
Using the renewal approach we prove Bernstein-like inequalities for additive functionals of geometrically ergodic Markov chains, thus obtaining counterparts of inequalities for sums of independent random variables. The coefficient in the sub-Gaussian part of our estimate is the asymptotic variance of the additive functional, i.e. the variance of the limiting Gaussian variable in the Central Limit Theorem for Markov chains. This refines earlier results by R. Adamczak and W. Bednorz, which were obtained under the additional assumption of strong aperiodicity of the chain.
From multitype branching Brownian motions to branching Markov additive processes
University of Chinese Academy of Sciences（中国科学院大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；LMAM School of Mathematical Sciences & Center for Statistical Science, Peking University（北京大学数学学院（LMAM）及统计科学中心）；School of Mathematical Sciences, Beijing University of Posts and Telecommunications（北京邮电大学数学科学学院）
We revisit functional central limit theorems for additive functionals of ergodic Markov diffusion processes. Translated in the language of partial differential equations of evolution, they appear as diffusion limits in the asymptotic analysis of Fokker-Planck type equations. We focus on the square integrable framework, and we provide tractable conditions on the infinitesimal generator, including degenerate or anomalously slow diffusions. We take advantage on recent developments in the study of the trend to the equilibrium of ergodic diffusions. We discuss examples and formulate open problems.
We provide results relating to the integrability, uniform integrability and local integrability of exponential MAPs, which are natural extensions of exponential Levy models. Then, we use Mellin transform and partial integro-differential equation methods to value European options under a such a model. Finally, a comparison is made between the price of a European call option and that of an Asian call option.
Previous authors have considered optimal stopping problems driven by the running maximum of a spectrally negative Lévy process $X$, as well as of a one-dimensional diffusion. Many of the aforementioned results are either implicitly or explicitly dependent on Peskir's maximality principle. In this article, we are interested in understanding how some of the main ideas from these previous works can be brought into the setting of problems driven by the maximum of a class of Markov additive processes (more precisely Markov modulated Lévy processes). Similarly to previous works in the Lévy setting, …
For a Harris ergodic Markov chain $(X_n)_{n\ge 0}$, on a general state space, started from the so called small measure or from the stationary distribution we provide optimal estimates for Orlicz norms of sums $\sum_{i=0}^τf(X_i)$, where $τ$ is the first regeneration time of the chain. The estimates are expressed in terms of other Orlicz norms of the function $f$ (wrt the stationary distribution) and the regeneration time $τ$ (wrt the small measure). We provide applications to tail estimates for additive functionals of the chain $(X_n)$ generated by unbounded functions as well as to classical l…
In this paper we consider an additive functional of an observable $V(x)$ of a Markov jump process. We assume that the law of the expected jump time $t(x)$ under the invariant probability measure $π$ of the skeleton chain belongs to the domain of attraction of a subordinator. Then, the scaled limit of the functional is a Mittag-Leffler proces, provided that $Ψ(x):=V(x)t(x)$ is square integrable w.r.t. $π$. When the law of $Ψ(x)$ belongs to a domain of attraction of a stable law the resulting process can be described by a composition of a stable process and the inverse of a subordinator and thes…
We derive the Markov-modulated generalized Ornstein-Uhlenbeck process by embedding a Markov-modulated random recurrence equation in continuous time. The obtained process turns out to be the unique solution of a certain stochastic differential equation driven by a bivariate Markov-additive process. We present this stochastic differential equation as well as its solution explicitely in terms of the driving Markov-additive process. Moreover, we give necessary and sufficient conditions for strict stationarity of the Markov-modulated generalized Ornstein-Uhlenbeck process, and prove that its statio…
In this paper we consider the first passage process of a spectrally negative Markov additive process (MAP). The law of this process is uniquely characterized by a certain matrix function, which plays a crucial role in fluctuation theory. We show how to identify this matrix using the theory of Jordan chains associated with analytic matrix functions. Importantly, our result also provides us with a technique, which can be used to derive various further identities. We then proceed to show how to compute the stationary distribution associated with a one-sided reflected (at zero) MAP for both the sp…
In this paper we solve the exit problems for an one-sided Markov additive process (MAP) which is exponentially killed with a bivariate killing intensity $ω(\cdot,\cdot)$ dependent on the present level of the process and the present state of the environment. Moreover, we analyze respective resolvents. All identities are given in terms of new generalizations of classical scale matrices for the MAP. We also remark on a number of applications of the obtained identities to (controlled) insurance risk processes. In particular, we show that our results can be applied to the so-called Omega model, whe…
We consider additive functionals of Markov processes in continuous time with general (metric) state spaces. We derive concentration bounds for their exponential moments and moments of finite order. Applications include diffusions, interacting particle systems and random walks. The method is based on coupling estimates and not spectral theory, hence reversibility is not needed. We bound the exponential moments(or the moments of finite order) in terms of a so-called coupled function difference, which in turn is estimated using the generalized coupling time. Along the way we prove a general relat…
IIn this paper we provide predictable and chaotic representations for Itô-Markov additive processes $X$. Such a process is governed by a finite-state CTMC $J$ which allows one to modify the parameters of the Itô-jump process (in so-called regime switching manner). In addition, the transition of $J$ triggers the jump of $X$ distributed depending on the states of $J$ just prior to the transition. This family of processes includes Markov modulated Itô-Lévy processes and Markov additive processes. The derived chaotic representation of a square-integrable random variable is given as a sum of stocha…
We consider a Markov additive process with a finite phase space and study its path decompositions at the times of extrema, first passage and last exit. For these three families of times we establish splitting conditional on the phase, and provide various relations between the laws of post- and pre-splitting processes using time reversal. These results offer valuable insight into behavior of the process, and while being structurally similar to the Lévy process case, they demonstrate various new features. As an application we formulate the Wiener-Hopf factorization, where time is counted in each…
A Markov Additive Process is a bi-variate Markov process $(ξ,J)=\big((ξ_t,J_t),t\geq0\big)$ which should be thought of as a multi-type Lévy process: the second component $J$ is a Markov chain on a finite space $\{1,\ldots,K\}$, and the first component $ξ$ behaves locally as a Lévy process, with local dynamics depending on $J$. In the subordinator-like case where $ξ$ is nondecreasing, we establish several results concerning the moments of $ξ$ and of its exponential functional $I_ξ=\int_{0}^{\infty} e^{-ξ_t}\mathrm dt,$ extending the work of Carmona et al., and Bertoin and Yor. We then apply the…
Motivated by various applications, we describe the scaling limits of bivariate Markov chains $(X,J)$ on $\mathbb Z_+ \times \{1,\ldots,κ\}$ where $X$ can be viewed as a position marginal and $\{1,\ldots,κ\}$ is a set of $κ$ types. The chain starts from an initial value $(n,i)\in \mathbb Z_+ \times \{1,\ldots,κ\}$, with $i$ fixed and $n \rightarrow \infty$, and typically we will assume that the macroscopic jumps of the marginal $X$ are rare, i.e. arrive with a probability proportional to a negative power of the current state. We also assume that $X$ is non-increasing. We then observe different …
This paper solves exit problems for spectrally negative Markov additive processes and their reflections. A so-called scale matrix, which is a generalization of the scale function of a spectrally negative \levy process, plays a central role in the study of exit problems. Existence of the scale matrix was shown in Thm. 3 of Kyprianou and Palmowski (2008). We provide a probabilistic construction of the scale matrix, and identify the transform. In addition, we generalize to the MAP setting the relation between the scale function and the excursion (height) measure. The main technique is based on th…
This paper develops a systematic treatment of monotonicity-based pathwise dualities for Markov processes taking values in partially ordered sets. We show that every Markov process that takes values in a finite partially ordered set and whose generator can be represented in monotone maps has a pathwise dual process. In the special setting of attractive spin systems this has been discovered earlier by Gray. We show that the dual simplifies a lot when the state space is a lattice (in the order-theoretic meaning of the word) and all monotone maps satisfy an additivity condition. This leads to a un…
Finitely-additive measures on the asymptotic foliations of a Markov compactum
建立了平面上平移流时间平均的渐近展开式。这一结果推广了A.Zorich和G.Forni的早期工作，得到了平移流的极限定理。的论点，接近的精神，G.Forni，使用的近似遍历积分的完整不变的Hoelder上循环的轨迹的流量。完整不变的Hoelder上循环空间是有限维的，并给出了一个明确的结构。首先，S.Ito和A. M. Vershik给出了唯一遍历平移流的一个符号表示，然后，利用马氏紧集的渐近叶理上的一族有限可加复值完整性-不变测度构造了上圈空间.
This paper proposes additional material to the main statements of [HL13] which are are recalled in Section~2. In particular an application of [Theorem~2.2,HL13] to Renewal Markov Processes is provided in Section~4 and a detailed checking of the assumptions of [Theorem~2.2,HL13] for the joint distribution of local times of a finite jump process is reported in Section~5. A uniform version of [Theorem~2.2,HL13] with respect to a compact set of transition matrices is given in Section~6 (see [Remark 2.4,HL13]). The basic material on the semigroup of Fourier matrices and the spectral approach used i…
