Large deviation results are given for a class of perturbed nonhomogeneous Markov chains on finite state space which formally includes some stochastic optimization algorithms. Specifically, let {P_n} be a sequence of transition matrices on a finite state space which converge to a limit transition matrix P. Let {X_n} be the associated nonhomogeneous Markov chain where P_n controls movement from time n-1 to n. The main statements are a large deviation principle and bounds for additive functionals of the nonhomogeneous process under some regularity conditions. In particular, when P is reducible, three regimes that depend on the decay of certain ``connection'' P_n probabilities are identified. Roughly, if the decay is too slow, too fast or in an intermediate range, the large deviation behavior is trivial, the same as the time-homogeneous chain run with P or nontrivial and involving the decay rates. Examples of anomalous behaviors are also given when the approach P_n\to P is irregular. Results in the intermediate regime apply to geometrically fast running optimizations, and to some issues in glassy physics.
Let $(ξ_n)_{n=0}^\infty$ be a nonhomogeneous Markov chain taking values from finite state-space of $\mathbf{X}=\{1,2,\ldots,b\}$. In this paper, we will study the generalized entropy ergodic theorem with almost-everywhere and $\mathcal{L}_1$ convergence for nonhomogeneous Markov chains, which generalizes the corresponding classical results for the Markov chains.
In this paper, we obtain new drift-type conditions for nonexplosion and explosion for nonhomogeneous Markov pure jump processes in Borel state spaces. The conditions are sharp; e.g., the one for nonexplosion is necessary if the state space is in addition locally compact and the $Q$-function satisfies weak Feller-type and local boundedness conditions. We comment on the relations of our conditions with the existing ones in the literature, and demonstrate some possible applications.
We prove that moderate deviations for empirical measures for countable nonhomogeneous Markov chains hold under the assumption of uniform convergence of transition probability matrices for countable nonhomogeneous Markov chains in Cesàro sense.
We consider finite-state time-nonhomogeneous Markov chains where the probability of moving from state $i$ to state $j\neq i$ at time $n$ is $G(i,j)/n^ζ$ for a ``generator'' matrix $G$ and strength parameter $ζ>0$. In these chains, as time grows, the positions are less and less likely to change, and so form simple models of age-dependent time-reinforcing behaviors. These chains, however, exhibit some different, perhaps unexpected, asymptotic occupation laws depending on parameters. Although on the one hand it is shown that the asymptotic position converges to a point-mixture for all $ζ>0$, on t…
In the present paper we investigate the $L_1$-weak ergodicity of nonhomogeneous discrete Markov processes with general state spaces. Note that the $L_1$-weak ergodicity is weaker than well-known weak ergodicity. We provide a necessary and sufficient condition for such processes to satisfy the $L_1$-weak ergodicity. Moreover, we apply the obtained results to establish $L_1$-weak ergodicity of discrete time quadratic stochastic processes. As an application of the main result, certain concrete examples are also provided.
This paper is about the existence and regularity of the transition probability matrix of a nonhomogeneous continuous-time Markov process with a countable state space. A standard approach to prove the existence of such a transition matrix is to begin with a continuous (in t) and conservative matrix Q(t)=[q_{ij}(t)] of nonhomogeneous transition rates q_{ij}(t), and use it to construct the transition probability matrix. Here we obtain the same result except that the q_{ij}(t) are only required to satisfy a mild measurability condition, and Q(t) may not be conservative. Moreover, the resulting tra…
We consider triangular arrays of Markov chains that converge weakly to a diffusion process. Second order Edgeworth type expansions for transition densities are proved. The paper differs from recent results in two respects. We allow nonhomogeneous diffusion limits and we treat transition densities with time lag converging to zero. Small time asymptotics are motivated by statistical applications and by resulting approximations for the joint density of diffusion values at an increasing grid of points.
We present a solution of a class of network utility maximization (NUM) problems using minimal communication. The constraints of the problem are inspired less by TCP-like congestion control but by problems in the area of internet of things and related areas in which the need arises to bring the behavior of a large group of agents to a social optimum. The approach uses only intermittent feedback, no inter-agent communication, and no common clock. The proposed algorithm is a combination of the classical AIMD algorithm in conjunction with a simple probabilistic rule for the agents to respond to a …
Our purpose is to prove central limit theorem for countable nonhomogeneous Markov chain under the condition of uniform convergence of transition probability matrices for countable nonhomogeneous Markov chain in Cesàro sense. Furthermore, we obtain a corresponding moderate deviation theorem for countable nonhomogeneous Markov chain by Gärtner-Ellis theorem and exponential equivalent method.
It is known that the Dobrushin's ergodicity coefficient is one of the effective tools to study a behavior of non-homogeneous Markov chains. In the present paper, we define such an ergodicity coefficient of a positive mapping defined on ordered Banach space with a base (OBSB), and study its properties. In terms of this coefficient we prove the equivalence uniform and weak ergodicities of homogeneous Markov chains. This extends earlier results obtained in case of von Neumann algebras. Such a result allowed to establish a category theorem for uniformly ergodic Markov operators. We find necessary …
Motivated by a model presented by S. Gudder, we study a quantum generalization of Markov chains and discuss the relation between these maps and open quantum random walks, a class of quantum channels described by S. Attal et al. We consider processes which are nonhomogeneous in time, i.e., at each time step, a possibly distinct evolution kernel. Inspired by a spectral technique described by L. Saloff-Coste and J. Zúñiga, we define a notion of ergodicity for nonhomogeneous quantum Markov chains and describe a criterion for ergodicity of such objects in terms of singular values. As a consequence …
We prove an analog of the classical Zero-One Law for both homogeneous and nonhomogeneous Markov chains (MC). Its almost precise formulation is simple: given any event $A$ from the tail $σ$-algebra of MC $(Z_n)$, for large $n$, with probability near one, the trajectories of the MC are in states $i$, where $P(A|Z_n=i)$ is either near $0$ or near $1$. A similar statement holds for the entrance $σ$-algebra, when $n$ tends to $-\infty$. To formulate this second result, we give detailed results on the existence of nonhomogeneous Markov chains indexed by $\mathbb Z_-$ or $\mathbb Z$ in both the finit…
PageRank is a ranking of the web pages that measures how often a given web page is visited by a random surfer on the web graph, for a simple model of web surfing. It seems realistic that PageRank may also have an influence on the behavior of web surfers. We propose here a simple model taking into account the mutual influence between web ranking and web surfing. Our ranking, the T-PageRank, is a nonlinear generalization of the PageRank. It is defined as the limit, if it exists, of some nonlinear iterates. A positive parameter T, the temperature, measures the confidence of the web surfer in the …
The paper is devoted to the estimation of the rate of of exponential convergence of nonhomogeneous queues exhibiting different types of ergodicity. The main tool of our study is the method, which was proposed by the second author in the late 1980-s and was subsequently extended and developed in different directions in a series of joint papers by the authors of the present paper. The method originated from the idea of Gnedenko and Makarov to employ the logarithmic norm of a matrix to the study of the problem of stability of nonhomogeneous Markov chains. In the present paper, we apply the method…
We extend in two directions our previous results about the sampling and the empirical measures of immortal branching Markov processes. Direct applications to molecular biology are rigorous estimates of the mutation rates of polymerase chain reactions from uniform samples of the population after the reaction. First, we consider nonhomogeneous processes, which are more adapted to real reactions. Second, recalling that the first moment estimator is analytically known only in the infinite population limit, we provide rigorous confidence intervals for this estimator that are valid for any finite po…
This paper describes the structure of solutions to Kolmogorov's equations for nonhomogeneous jump Markov processes and applications of these results to control of jump stochastic systems. These equations were studied by Feller (1940), who clarified in 1945 in the errata to that paper that some of its results covered only nonexplosive Markov processes. In this work, which is largely of a survey nature, the case of explosive processes is also considered. This paper is based on the invited talk presented by the authors at the conference "Chebyshev-200", and it describes the results of their joine…
In this paper we are concerned with hitting times of a family of density-dependent Markov chains. A moderate deviation principle of the hitting time is given. The proof of the main theorem relies heavily on moderate deviations of density-dependent Markov chains given in \cite{Xue2021} and upper bounds of large deviations of Markov processes given in \cite{Dupuis1991}. An analogue moderate deviation of the hitting time of the diffusion approximation of the density-dependent Markov chain introduced in \cite{Ethier1986} is also given.
In this paper, we study small noise asymptotics of Markov-modulated diffusion processes in the regime that the modulating Markov chain is rapidly switching. We prove the joint sample-path large deviations principle for the Markov-modulated diffusion process and the occupation measure of the Markov chain (which evidently also yields the large deviations principle for each of them separately by applying the contraction principle). The structure of the proof is such that we first prove exponential tightness, and then establish a local large deviations principle (where the latter part is split int…
The density-dependent Markov chain (DDMC) introduced in \cite{Kurtz1978} is a continuous time Markov process applied in fields such as epidemics, chemical reactions and so on. In this paper, we give moderate deviation principles of paths of DDMC under some generally satisfied assumptions. The proofs for the lower and upper bounds of our main result utilize an exponential martingale and a generalized version of Girsanov's theorem. The exponential martingale is defined according to the generator of DDMC.
Consider a sequence of continuous-time Markov chains $(X^{(n)}_t:t\ge 0)$ evolving on a fixed finite state space $V$. Let $I_n$ be the level two large deviations rate functional for $X^{(n)}_t$, as $t\to\infty$. Under a hypothesis on the jump rates, we prove that $I_n$ can be written as $I_n = I^{(0)} \,+\, \sum_{1\le p\le q} (1/θ^{(p)}_n) \, I^{(p)}$ for some rate functionals $I^{(p)}$. The weights $θ^{(p)}_n$ correspond to the time-scales at which the sequence of Markov chains $X^{(n)}_t$ exhibit a metastable behavior, and the zero level sets of the rate functionals $I^{(p)}$ identify the me…
We prove that the long term distribution of the queue length process in an ergodic generalised Jackson network obeys the Large Deviation Principle with a deviation function given by the quasipotential. The latter is related to the unique long term idempotent distribution, which is also a stationary idempotent distribution, of the large deviation limit of the queue length processes. The proof draws on developments in queueing network stability and idempotent probability.
We provide deviation inequalities for properly normalized sums of bifurcating Markov chains on Galton-Watson tree. These processes are extension of bifurcating Markov chains (which was introduced by Guyon to detect cellular aging from cell lineage) in case the index set is a binary Galton-Watson process. As application, we derive deviation inequalities for the least-squares estimator of autoregressive parameters of bifurcating autoregressive processes with missing data. These processes allow, in case of cell division, to take into account the cell's death. The results are obtained under an uni…
In this paper we derive non asymptotic deviation bounds for $$¶_ν(|\frac 1t \int_0^t V(X_s) ds - \int V dμ| \geq R)$$ where $X$ is a $μ$ stationary and ergodic Markov process and $V$ is some $μ$ integrable function. These bounds are obtained under various moments assumptions for $V$, and various regularity assumptions for $μ$. Regularity means here that $μ$ may satisfy various functional inequalities (F-Sobolev, generalized Poincaré etc...).
We consider Markov chains which are polynomially mixing, in a weak sense expressed in terms of the space of functions on which the mixing speed is controlled. In this context, we prove polynomial large and moderate deviations inequalities. These inequalities can be applied in various natural situations coming from probability theory or dynamical systems. Finally, we discuss examples from these various settings showing that our inequalities are sharp.
We study the Hamilton-Jacobi equation f - lambda Hf = h, where H f = e^{-f}Ae^f and where A is an operator that corresponds to a well-posed martingale problem. We identify an operator that gives viscosity solutions to the Hamilton-Jacobi equation, and which can therefore be interpreted as the resolvent of H. The operator is given in terms of optimization problem where the running cost is a path-space relative entropy. Finally, we use the resolvents to give a new proof of the abstract large deviation result of Feng and Kurtz.
We consider a continuous time Markov chain on a countable state space and prove a joint large deviation principle for the empirical measure and the empirical flow, which accounts for the total number of jumps between pairs of states. We give a direct proof using tilting and an indirect one by contraction from the empirical process.
For Markov processes evolving on multiple time-scales a combination of large component scalings and averaging of rapid fluctuations can lead to useful limits for model approximation. A general approach to proving a law of large numbers to a deterministic limit and a central limit theorem around it have already been proven in [16] and [17]. We present here a general approach to proving a large deviation principle in path space for such multi-scale Markov processes. Motivated by models arising in systems biology, we apply these large deviation results to general chemical reaction systems which e…
A large deviations principle is established for the joint law of the empirical measure and the flow measure of a renewal Markov process on a finite graph. We do not assume any bound on the arrival times, allowing heavy tailed distributions. In particular, the rate functional is in general degenerate (it has a nontrivial set of zeros) and not strictly convex. These features show a behavior highly different from what one may guess with a heuristic Donsker-Varadhan analysis of the problem.
First, under a geometric ergodicity assumption, we provide some limit theorems and some probability inequalities for the bifurcating Markov chains (BMC). The BMC model was introduced by Guyon to detect cellular aging from cell lineage, and our aim is thus to complete his asymptotic results. The deviation inequalities are then applied to derive first result on the moderate deviation principle (MDP) for a functional of the BMC with a restricted range of speed, but with a function which can be unbounded. Next, under a uniform geometric ergodicity assumption, we provide deviation inequalities for …
We prove pathwise large-deviation principles of switching Markov processes by exploiting the connection to associated Hamilton-Jacobi equations, following Jin Feng's and Thomas Kurtz's method. In the limit that we consider, we show how the large-deviation problem in path-space reduces to a spectral problem of finding principal eigenvalues. The large-deviation rate functions are given in action-integral form. As an application, we demonstrate how macroscopic transport properties of stochastic models of molecular motors can be deduced from an associated principal-eigenvalue problem. The precise …
Here we propose the Donsker-Varadhan-type compactness conditions and prove the joint large deviation principle for the empirical measure and empirical flow of Markov renewal processes (semi-Markov processes) with a countable state space, generalizing the relevant results for continuous-time Markov chains with a countable state space obtained in [Ann. Inst. H. Poincaré Probab. Statist. 51, 867-900 (2015)] and [Stoch. Proc. Appl. 125, 2786-2819 (2015)], as well as the relevant results for Markov renewal processes with a finite state space obtained in [Adv. Appl. Probab. 48, 648-671 (2016)]. In p…
Our aim is to unify and extend the large deviation upper and lower bounds for the occupation times of a Markov process with $L_2$ semigroups under minimal conditions on the state space and the process trajectories; for example, no strong Markov property is needed. The methods used here apply in both continuous and discrete time. We present the proofs for continuous time only because of the inherent technical difficulties in that situation; the proofs can be adapted for discrete time in a straightforward manner.
The main purpose of this article is to establish moderate deviation principles for additive functionals of bifurcating Markov chains. Bifurcating Markov chains are a class of processes which are indexed by a regular binary tree. They can be seen as the models which represent the evolution of a trait along a population where each individual has two offsprings. Unlike the previous results of Bitseki, Djellout \& Guillin (2014), we consider here the case of functions which depend only on one variable. So, mainly inspired by the recent works of Bitseki \& Delmas (2020) about the central limit theo…
This paper studies the subgeometric convergence of the stationary distribution in taking the infinite-level limit of a finite-level M/G/1-type Markov chain, that is, in letting the upper boundary level go to infinity. This study is performed through the fundamental deviation matrix, which is a block-decomposition-friendly solution for the Poisson equation of the deviation matrix. The fundamental deviation matrix yields a difference formula for the respective stationary distributions of the finite-level chain and the corresponding infinite-level chain. The difference formula plays a crucial rol…
The large deviations principle for the empirical measure for both continuous and discrete time Markov processes is well known. Various expressions are available for the rate function, but these expressions are usually as the solution to a variational problem, and in this sense not explicit. An interesting class of continuous time, reversible processes was identified in the original work of Donsker and Varadhan for which an explicit expression is possible. While this class includes many (reversible) processes of interest, it excludes the case of continuous time pure jump processes, such as a re…
Hawkes process is a class of simple point processes that is self-exciting and has clustering effect. The intensity of this point process depends on its entire past history. It has wide applications in finance, neuroscience and many other fields. In this paper, we study the large deviations for nonlinear Hawkes processes. The large deviations for linear Hawkes processes has been studied by Bordenave and Torrisi. In this paper, we prove first a large deviation principle for a special class of nonlinear Hawkes processes, that is, a Markovian Hawkes process with nonlinear rate and exponential exci…
In this paper, the large deviations on trajectory level for ergodic Markov processes are studied. These processes take values in the non-negative quadrant of the two dimension lattice and are concentrated on step-wise functions. The rates of jumps towards the axes (jump down) depend on the position of the process -- the higher the position, the greater the rate. The rates of jumps going in the same direction as the axes (jump up) are constants. Therefore the processes are ergodic. The large deviations are studied under equal scalings of both space and time. The scaled versions of the processes…
We prove a large deviation principle on path space for a class of discrete time Markov processes whose state space is the intersection of a regular domain $Ł\subset \R^d$ with some lattice of spacing $\e$. Transitions from $x$ to $y$ are allowed if $\e^{-1}(x-y)\in \D$ for some fixed set of vectors $\D$. The transition probabilities $p_\e(t,x,y)$, which themselves depend on $\e$, are allowed to depend on the starting point $x$ and the time $t$ in a sufficiently regular way, except near the boundaries, where some singular behaviour is allowed. The rate function is identified as an action functi…
In this paper we consider two related stochastic models. The first one is a branching system consisting of particles moving according to a Markov family in R^d and undergoing subcritical branching with a constant rate of V>0. New particles immigrate to the system according to a homogeneous space time Poisson random field. The second model is the superprocess corresponding to the branching particle system. We study rescaled occupation time process and the process of its fluctuations with very mild assumptions on the Markov family. In the general setting a functional central limit theorem as wel…
In this paper, we present new Poisson-type deviation inequalities for continuous-time Markov chains whose Wasserstein curvature or $Γ$-curvature is bounded below. Although these two curvatures are equivalent for Brownian motion on Riemannian manifolds, they are not comparable in discrete settings and yield different deviation bounds. In the case of birth--death processes, we provide some conditions on the transition rates of the associated generator for such curvatures to be bounded below and we extend the deviation inequalities established [Ané, C. and Ledoux, M. On logarithmic Sobolev inequa…
For ${1/2}<α<1$, we propose the MDP analysis for family $$ S^α_n=\frac{1}{n^α}\sum_{i=1}^nH(X_{i-1}), n\ge 1, $$ where $(X_n)_{n\ge 0}$ be a homogeneous ergodic Markov chain, $X_n\in \mathbb{R}^d$, when the spectrum of operator $P_x$ is continuous. The vector-valued function $H$ is not assumed to be bounded but the Lipschitz continuity of $H$ is required. The main helpful tools in our approach are Poisson's equation and Stochastic Exponential; the first enables to replace the original family by $\frac{1}{n^α}M_n$ with a martingale $M_n$ while the second to avoid the direct Laplace transform an…
For ${1/2}<α<1$, we propose the MDP analysis for family $$ S^α_n=\frac{1}{n^α}\sum_{i=1}^nH(X_{i-1}), n\ge 1, $$ where $(X_n)_{n\ge 0}$ be a homogeneous ergodic Markov chain, $X_n\in \mathbb{R}^d$, when the spectrum of operator $P_x$ is continuous. The vector-valued function $H$ is not assumed to be bounded but the Lipschitz continuity of $H$ is required. The main helpful tools in our approach are Poisson equation and Stochastic Exponential; the first enables to replace the original family by $\frac{1}{n^α}M_n$ with a martingale $M_n$ while the second to avoid the direct Laplace transform anal…
Large deviations related to Dynkin--Lamperti arcsine laws for last visit times of Markov processes
Faculty of Arts and Sciences, Kyoto Institute of Technology（京都工艺纤维大学艺术与科学学院）；Department of Mathematics, Graduate School of Science, The University of Osaka（大阪大学理学研究科数学系）；Department of Systems Innovation, Graduate School of Engineering Science, The University of Osaka（大阪大学工学研究科系统创新系）
For a Markov process associated with a diffusion type Dirichlet form an upper bound is shown for the law of the finite dimensional distributions of the process. Under some more assumptions on the underlaying space this is also shown for the law of the Markov process itself. In the last section we want to give an application to the Wasserstein diffusion.
A large deviation principle is established for a two-scale stochastic system in which the slow component is a continuous process given by a small noise finite dimensional Itô stochastic differential equation, and the fast component is a finite state pure jump process. Previous works have considered settings where the coupling between the components is weak in a certain sense. In the current work we study a fully coupled system in which the drift and diffusion coefficient of the slow component and the jump intensity function and jump distribution of the fast process depend on the states of both…
We compute the Hamiltonian and Lagrangian associated to the large deviations of the trajectory of the empirical distribution for independent Markov processes, and of the empirical measure for translation invariant interacting Markov processes. We treat both the case of jump processes (continuous-time Markov chains and interacting particle systems) as well as diffusion processes. For diffusion processes, the Lagrangian is a quadratic form of the deviation of the trajectory from the Kolmogorov forward equation. In all cases, the Lagrangian can be interpreted as a relative entropy or relative ent…
Many networking-related settings can be modeled by Markov-modulated infinite-server systems. In such models, the customers' arrival rates and service rates are modulated by a Markovian background process, additionally, there are infinitely many servers (and consequently the resulting model is often used as a proxy for the corresponding many-server model). The Markov-modulated infinite-server model hardly allows any explicit analysis, apart from results in terms of systems of (ordinary or partial) differential equations for the underlying probability generating functions, and recursions to obta…
We investigate the transportation cost-information inequalities for bifurcating Markov chains which are a class of processes indexed by binary tree. These processes provide models for cell growth when each individual in one generation gives birth to two offsprings in the next one. Transportation cost inequalities provide useful concentra-tion inequalities. We also study deviation inequalities for the empiri-cal means under relaxed assumptions on the Wasserstein contraction of the Markov kernels. Applications to bifurcating non linear autore-gressive processes are considered: deviation inequali…
We consider a class of Markov processes with resettings, where at random times, the Markov processes are restarted from a predetermined point or a region. These processes are frequently applied in physics, chemistry, biology, economics, and in population dynamics. In this paper we establish the local large deviation principle (LLDP) for the Wiener processes with random resettings, where the resettings occur at the arrival time of a Poisson process. Here, at each resetting time, a new resetting point is selected at random, according to a conditional distribution.
The purpose of this paper is to ensure the conditions of Gärtner-Ellis Theorem for evaluations of the empirical measure. We show that up-to-date conditions for ensuring the convergence to a quasi-stationary distribution can be applied efficiently. By this mean, we are able to prove Large Deviation results even with a conditioning that the process is not extinct at the end of the evaluation. The domain on which these Large Deviation results apply is implicitly given by the range of penalization for which one can prove the above-mentioned results of quasi-stationarity. We propose a way to relate…
