Galina A. Zverkina
We give strong bounds for the rate of convergence of the regenerative process distribution to the stationary distribution in the total variation metric. These bounds are obtained by using coupling method. We propose this method for obtaining such bounds for the queueing regenerative processes.
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We propose a new modification of the coupling method for renewal process in continuous time. We call this modification "the stationary coupling method", and construct it primarily to obtain the bounds for convergence rate of the distribution of the regenerative processes in the total variation metrics. At the same time this modification of the coupling method demonstrates an improvement of the classical result of polynomial convergence rate of the distribution of the regenerative process -- in the case of a heavy tail. This paper is an extension of my preprint "On strong bounds of rate of conv…
About quasi-renewal processes and quasi-regenerative processes
本文探讨了拟更新与拟再生过程的概念，并提出了一种在过程遍历条件下，获取再生与拟再生过程分布向平稳分布收敛速率上界的方法。。
The standard functional central limit theorem for a renewal process with finite mean and variance, results in a Brownian motion limit. This note shows how to obtain a Brownian bridge process by a direct procedure that does not involve conditioning. Several examples are also considered.
We consider regenerative processes with values in some Polish space. We define their ε-big excursions as excursions e such that f(e)>ε, where f is some given functional on the space of excursions which can be thought of as, e.g., the length or the height of e. We establish a general condition that guarantees the convergence of a sequence of regenerative processes involving the convergence of ε-big excursions and of their endpoints, for all εin a countable set whose closure contains 0. Finally, we provide various sufficient conditions on the excursion measures of this sequence for this general …
The paper investigates the convergence of the distribution of some non-regenerative reliability system with a warm reserve.
We identify general conditions under which regenerative processes with dependent cycles and cycle lengths are asymptotically independent. The result is applied to various models. In particular, independent Lévy processes with dependent secondary jumps at the origin (e.g., workloads of parallel M/G/1 queues with server vacations), the asymptotic performance of real-time status systems with multiple correlated sources measured by the stationary probability of an updated system and asymptotic results for clearing processes with dependent arrivals of inputs and clearings.
We review the theory of regenerative processes, which are processes that can be intuitively seen as comprising of i.i.d.\ cycles. Although we focus on the classical definition, we present a more general definition that allows for some form of dependence between two adjacent cycles, and mention two further extensions of the second definition. We mention the connection of regenerative processes to the single-server queue, to multi-server queues and more generally to Harris ergodic Markov chains and processes. In the main theorem, we pay some attention to the conditions under which a limiting dis…
This paper demonstrates a new regeneration processes technology making use of positive stable distributions. We study the asymptotic behavior of branching processes with a randomly controlled migration component. Using the new method, we confirm some known results and establish new limit theorems that hold in a more general setting.
The exponential upper bounds for the convergence rate of the distribution of restorable element with partially energized standby redundancy are founded, in the case when all working and repair times are bounded by exponential random variable (upper and lower), and working and repair times can by dependent. The bounds for the convergence rate of the availability factor are estimated. -------------------------------------- Polucheny ocenki sverhu dlya skorosti skhodimosti raspredeleniya sostoyaniya vosstanavlivaemogo ehlementa s tyoplym rezervom v sluchae, kogda vremena raboty i vosstanovleniya …
Renewal and Regenerative Processes: short review
这篇随机过程领域的简短综述聚焦马尔可夫、更新、再生及平稳过程，梳理核心结论与Thorisson的相关成果，明确其在多领域的应用价值，为后续研究提供基础。
State-dependent inverse-subordinator time changes of regenerative processes: Excursion structure and multiscale occupation-time limits
本文通过将不同独立子序分配给可测游程类，构造状态依赖逆子序时间变换的再生过程，并证明变换后的游程点过程仍为泊松过程，进而推导出多尺度联合占据时间极限定理。
We introduce regenerative tree growth processes as consistent families of random trees with n labelled leaves, n>=1, with a regenerative property at branch points. This framework includes growth processes for exchangeably labelled Markov branching trees, as well as non-exchangeable models such as the alpha-theta model, the alpha-gamma model and all restricted exchangeable models previously studied. Our main structural result is a representation of the growth rule by a sigma-finite dislocation measure kappa on the set of partitions of the natural numbers extending Bertoin's notion of exchangeab…
We investigate a family of discrete-time stationary processes defined by multiple stable integrals and renewal processes with infinite means. The model may exhibit behaviors of short-range or long-range dependence, respectively, depending on the parameters. The main contribution is to establish a phase transition in terms of the tail processes that characterize local clustering of extremes. Moreover, in the short-range dependence regime, the model provides an example where the extremal index is different from the candidate extremal index.
We derive an invariance principle for the lift to the rough path topology of stochastic processes with delayed regenerative increments under an optimal moment condition. An interesting feature of the result is the emergence of area anomaly, a correction term in the second level of the limiting rough path which is identified as the average stochastic area on a regeneration interval. A few applications include random walks in random environment and additive functionals of recurrent Markov chains. The result is formulated in the p-variation settings, where a rough Donsker Theorem is available und…
We analyze almost sure asymptotic behavior of extreme values of a regenerative process. We show that under certain conditions a properly centered and normalized running maximum of a regenerative process satisfies a law of the iterated logarithm for the $\limsup$ and a law of the triple logarithm for the $\liminf$. This complements a previously known result of Glasserman and Kou [Ann. Appl. Probab. 5(2) (1995), 424--445]. We apply our results to several queuing systems and a birth and death process.
We prove Berry-Esseen type rates of convergence for central limit theorems (CLTs) of regenerative processes which generalize previous results of Bolthausen under weaker moment assumptions. We then show how this general result can be applied to obtain rates of convergence for (1) CLTs for additive functionals of positive recurrent Markov chains under certain conditions on the strong mixing coefficients, and (2) annealed CLTs for certain ballistic random walks in random environments.
Using the regenerative scheme of Comets, Fernández and Ferrari (2002), we establish a functional central limit theorem (FCLT) for discrete time stochastic processes (chains) with summable memory decay. Furthermore, under stronger assumptions on the memory decay, we identify the limiting variance in terms of the process only. As applications, we define classes of binary autoregressive processes and power-law Ising chains for which the FCLT is fulfilled.
The ability to estimate the rate of convergence for the distributions of regenerative processes is in great demand. These processes are often encountered in queuing theory and in related problems. In some papers on regenerative processes, the coupling method and subsequent estimation of the convergence rate is used. In the presented small publication, we describe a general approach to using the coupling method for regenerative processes - both in the case of a polynomial and an exponential rate of convergence.
Regenerative processes for Poisson zero polytopes
本文研究了泊松零多面体的再生过程，证明其为伯努利流，并探讨了其在STIT划分过程中的应用。
Random spanning trees are among the most prominent determinantal point processes. We give four examples of random spanning trees on ladder-like graphs whose rungs form stationary renewal processes or regenerative processes of order two, respectively. Up to a trivial thinning with additional coin flips, for each of the first two examples the renewal processes exhaust the whole class of stationary regenerative (of order one) determinantal point processes. We also give an example of a regenerative process of order two that has no representation in terms of a random spanning tree. Our examples ill…
We show that a large class of stationary continuous-time regenerative processes are finitarily isomorphic to one another. The key is showing that any stationary renewal point process whose jump distribution is absolutely continuous with exponential tails is finitarily isomorphic to a Poisson point process. We further give simple necessary and sufficient conditions for a renewal point process to be finitarily isomorphic to a Poisson point process. This improves results and answers several questions of Soo and of Kosloff and Soo.
Consider a stochastic process $\mathfrak{X}$, regenerative at a state $x$ which is instantaneous and regular. Let $L$ be a regenerative local time for $\mathfrak{X}$ at $x$. Suppose furthermore that $\mathfrak{X}$ can be approximated by discrete time regenerative processes $\mathfrak{X}^n$ for which $x$ is accesible. We give conditions on $\mathfrak{X}$ and $\mathfrak{X}^n$ so that the naturally defined local time of $\mathfrak{X}^n$ converges weakly to $L$. This limit theorem generalizes previous invariance principles that have appeared in the literature.
Hermite processes are a class of self-similar processes with stationary increments. They often arise in limit theorems under long-range dependence. We derive new representations of Hermite processes with multiple Wiener-Itô integrals, whose integrands involve the local time of intersecting stationary stable regenerative sets. The proof relies on an approximation of regenerative sets and local times based on a scheme of random interval covering.
In queuing theory and related problems, it is very important to know the numerical characteristics of an investigated system - both in stationary and non-stationary modes. In some cases, such characteristics can be calculated, but this is possible for a limited number of real models. However, in most cases, it is possible to calculate or estimate the stationary values of the characteristics of the studied models. We can use linear Markov processes to model how the overwhelming majority of queuing systems (QS), reliability systems, and queuing networks (QNS) operate; the processes are regenerat…
We introduce a family of branch merging operations on continuum trees and show that Ford CRTs are distributionally invariant. This operation is new even in the special case of the Brownian CRT, which we explore in more detail. The operations are based on spinal decompositions and a regenerativity preserving merging procedure of $(α, θ)$-strings of beads, that is, random intervals $[0, L_{α, θ}]$ equipped with a random discrete measure $dL^{-1}$ arising in the limit of ordered $(α, θ)$-Chinese restaurant processes as introduced recently by Pitman and Winkel. Indeed, we iterate the branch mergin…
Estimates of the probability of a regenerative process reaching a high level
本文针对再生过程达到高水平的概率估计问题，借助带延迟的几何和的两个辅助结果建立了双侧估计，并给出了其在排队论排队长度随机过程中的应用示例。
Many regenerative arguments in stochastic processes use random times which are akin to stopping times, but which are determined by the future as well as the past behaviour of the process of interest. Such arguments based on "conditioning on the future" are usually developed in an ad-hoc way in the context of the application under consideration, thereby obscuring underlying structure. In this paper we give a simple, unified and more general treatment of such conditioning theory. We further give a number of novel applications to various particle system models, in particular to various flavours o…
Regenerative Rejection Sampling
本文提出再生拒绝采样算法，基于经典拒绝采样和MCMC方法，构建连续时间再生过程，其平稳分布与目标密度一致。该方法无需有限上界常数，收敛速度更快，实验显示其具有更低的自相关性和更快的混合速度。
We present a perfect simulation algorithm for stationary processes indexed by Z, with summable memory decay. Depending on the decay, we construct the process on finite or semi-infinite intervals, explicitly from an i.i.d. uniform sequence. Even though the process has infinite memory, its value at time 0 depends only on a finite, but random, number of these uniform variables. The algorithm is based on a recent regenerative construction of these measures by Ferrari, Maass, Mart{\'ı}nez and Ney. As applications, we discuss the perfect simulation of binary autoregressions and Markov chains on the …
Some, but not all processes of the form $M_t=\exp(-ξ_t)$ for a pure-jump subordinator $ξ$ with Laplace exponent $Φ$ arise as residual mass processes of particle 1 (tagged particle) in Bertoin's partition-valued exchangeable fragmentation processes. We introduce the notion of a Markovian embedding of $M=(M_t,t\ge 0)$ in a fragmentation process, and we show that for each $Φ$, there is a unique (in distribution) binary fragmentation process in which $M$ has a Markovian embedding. The identification of the Laplace exponent $Φ^*$ of its tagged particle process $M^*$ gives rise to a symmetrisation o…
A data-dependent DKW inequality for regenerative Markov chains
针对具有再生结构的马尔可夫链，提出数据依赖的Dvoretzky-Kiefer-Wolfowitz不等式，给出经验CDF的均匀置信带，并通过反演得到分位数函数的置信带，其宽度主项可直接从样本路径计算。
Results for convergence rates associated with renewal processes
本文通过耦合论证得到散布型间隔到达分布下更新方程解的幂律衰减速率，结合再生性质证明更新过程复发时间与更新计数过程补偿子的依分布收敛性，所有结果仅需矩存在的温和假设。
We prove regenerative properties for the linear Hawkes process under minimal assumptions on the transfer function, which may have unbounded support. These results are applicable to sliding window statistical estimators. We exploit independence in the Poisson cluster point process decomposition, and the regeneration times are not stopping times for the Hawkes process. The regeneration time is interpreted as the renewal time at zero of a M/G/infinity queue, which yields a formula for its Laplace transform. When the transfer function admits some exponential moments, we stochastically dominate the…
Given a two-sided real-valued Lévy process $(X_t)_{t \in \mathbb{R}}$, define processes $(L_t)_{t \in \mathbb{R}}$ and $(M_t)_{t \in \mathbb{R}}$ by $L_t := \sup\{h \in \mathbb{R} : h - α(t-s) \le X_s \text{ for all } s \le t\} = \inf\{X_s + α(t-s) : s \le t\}$, $t \in \mathbb{R}$, and $M_t := \sup \{ h \in \mathbb{R} : h - α|t-s| \leq X_s \text{ for all } s \in \mathbb{R} \} = \inf \{X_s + α|t-s| : s \in \mathbb{R}\}$, $t \in \mathbb{R}$. The corresponding contact sets are the random sets $\mathcal{H}_α:= \{ t \in \mathbb{R} : X_{t}\wedge X_{t-} = L_t\}$ and $\mathcal{Z}_α:= \{ t \in \mathbb{…
A new, conceptual proof approach for establishing the existence of regenerative space-time points for symmetric, translation invariant, finite-range interaction contact processes on survival is shown. The proof is elementary, complements the original one, and employs symmetry-based coupling arguments and a new consequence of convergence to equilibrium of the process in order to circumvent the original block construction.
We give a scheme of using the coupling method to obtain strong bounds for the convergence rate of the distribution of the backward renewal process in the total variation distance. This scheme can be applied to a wide class of regenerative processes in queuing theory.
We prove limit theorems of an entirely new type for certain long memory regularly varying stationary infinitely divisible random processes. These theorems involve multiple phase transitions governed by how long the memory is. Apart from one regime, our results exhibit limits that are not among the classical extreme value distributions. Restricted to the one-dimensional case, the distributions we obtain interpolate, in the appropriate parameter range, the $α$-Fréchet distribution and the skewed $α$-stable distribution. In general, the limit is a new family of stationary and self-similar random …
Let $\mathbb{X}=(\mathbb{X}_t)_{t\geq 0}$ be the subdiffusive process defined, for any $t\geq 0$, by $ \mathbb{X}_t = X_{\ell_t}$ where $X=(X_t)_{t\geq 0}$ is a Lévy process and $\ell_t=\inf \{s>0;\: \mathcal{K}_s>t \}$ with $\mathcal{K}=(\mathcal{K}_t)_{t\geq 0}$ a subordinator independent of $X$. We start by developing a composite Wiener-Hopf factorization to characterize the law of the pair $(\mathbb{T}_a^{(\mathcal{b})}, (\mathbb{X} - \mathcal{b})_{\mathbb{T}_a^{(\mathcal{b})}})$ where \begin{equation*} \mathbb{T}_a^{(\mathcal{b})} = \inf \{t>0;\: \mathbb{X}_t > a+ \mathcal{b}_t \} \end{eq…
Weak convergence of the integral of semi-Markov processes
本文研究再生过程与马尔可夫更新过程的弱渐近性质，推导相关更新型与遍历型结果，以此为框架证明半马尔可夫过程的积分在温和条件下弱收敛于缩放布朗运动，涵盖经典广义电报过程的弱收敛结论。
We consider a driven tagged particle in a symmetric exclusion process on Z with a removal rule. In this process, untagged particles are removed once they jump to the left of the tagged particle. We investigate the behavior of the displacement of the tagged particle and prove limit theorems of it: an (annealed) law of large numbers, a central limit theorem, and a large deviation principle. We also characterize a class of ergodic measures for the environment process. Our approach is based on analyzing two auxiliary processes with associated martingales and a regenerative structure.
In this paper, we prove the exponential convergence of the non-stationary moment of a random variable that defines the virtual waiting time in the mass service system M\G\1\ $\infty $, where the distribution of the service time satisfies the Kramer condition and system load factor $ρ< 1$, to the value of the virtual waiting time in the stationary mode in this system. We consider the virtual waiting time as a regenerative process, which regenerative periods is the sum of busy period and period of time, when the system is free. We show, that the busy period in such system satisfies the Kramer co…
We study the long-time behaviour of a class of piecewise-deterministic Markov processes which are an extension of some recent works. These $d$-dimensional processes, d>=1, can especially be used to model the motion of a bacterium in presence of a chemo-attractant, with parameters depending both on the position and the velocity of the bacterium. Using the method of Meyn and Tweedie, we show that under some good assumptions on the parameters of the model, such a process converges exponentially fast towards its invariant measure. We also establish the existence of exponential moments of the invar…
In this paper we consider heavy tailed Markov renewal processes and we prove that, suitably renormalised, they converge in law towards the $\ga$-stable regenerative set. We then apply these results to the strip wetting model which is a random walk $S$ constrained above a wall and rewarded or penalized when it hits the strip $[0,\infty) \times [0,a]$ where $a$ is a given positive number. The convergence result that we establish allows to characterize the scaling limit of this process at criticality.
A dependent complex degrading system with non-periodic inspection times
本文研究多退化过程相依且随机起始的系统，用伽马增量和非齐次泊松过程建模，推导可靠度及其界，并提出基于检测信息的动态检测策略，结合半再生过程分析维护成本。
We study systems of simple point processes that admit stochastic intensities. We represent these point processes as thinnings of Poisson measures and are interested in a convergence result of such systems. This result states that, if the stochastic intensities of the limit point processes are independent of the underlying Poisson measures, the convergence in distribution in Skorohod topology of the stochastic intensities implies the same convergence for the point processes.
Martingale techniques play a fundamental role in the analysis of point processes on the real line. In particular, the compensator of a point process uniquely determines and is determined by its distribution, and an explicit formula involving conditional interarrival distributions is well-known. In two dimensions there are many possible definitions of a point process compensator and we focus here on the one that has been the most useful in practice: the so-called *-compensator. Although existence of the *-compensator is well understood, in general it does not determine the law of the point proc…
We characterise the convergence of a certain class of discrete time Markov processes toward locally Feller processes in terms of convergence of associated operators. The theory of locally Feller processes is applied to Lévy-type processes in order to obtain convergence results on discrete and continuous time indexed processes, simulation methods and Euler schemes. We also apply the same theory to a slightly different situation, in order to get results of convergence of diffusions or random walks toward singular diffusions. As a consequence we deduce the convergence of random walks in random me…
Necessary and sufficient conditions for convergence in distribution of first-rare-event times and convergence in Skorokhod J-topology of first-rare-event-time processes for perturbed semi-Markov processes with finite phase space are obtained.
We consider symmetric processes of pure jump type. We prove local estimates on the probability of exiting balls, the Hölder continuity of harmonic functions and of heat kernels, and convergence of a sequence of such processes.
We obtain the lower bounds for ergodic convergence rates, including spectral gaps and convergence rates in strong ergodicity for time-changed symmetric Lévy processes by using harmonic function and reversible measure. As direct applications, explicit sufficient conditions for exponential and strong ergodicity are given. Some examples are also presented.
In a previous paper we have constructed a family of processes, starting from a set of independent standard Poisson processes, that has realizations that converge almost surely to the Brownian sheet, uniformly in the unit square. Now, a rate of convergence from these processes to Brownian sheet is given.
The paper investigates uniform convergence of wavelet expansions of Gaussian random processes. The convergence is obtained under simple general conditions on processes and wavelets which can be easily verified. Applications of the developed technique are shown for several classes of stochastic processes. In particular, the main theorem is adjusted to the fractional Brownian motion case. New results on the rate of convergence of the wavelet expansions in the space $C([0,T])$ are also presented.
In this paper, an integration by parts formula was derived for jump processes on Hilbert spaces. Using this formula, we investigated derivative formula and exponential ergodicity for nonlinear SPDEs driven by purely jump processes.
New results on uniform convergence in probability for expansions of Gaussian random processes using compactly supported wavelets are given. The main result is valid for general classes of nonstationary processes. An application of the obtained results to stationary processes is also presented. It is shown that the convergence rate of the expansions is exponential.
The paper characterizes uniform convergence rate for general classes of wavelet expansions of stationary Gaussian random processes. The convergence in probability is considered.
We consider the empirical measures of multi-type voter models with mutation on large finite sets, and prove their weak atomic convergence in the sense of Ethier and Kurtz (1994) toward a Fleming-Viot process. Convergence in the weak atomic topology is strong enough to answer a line of inquiry raised by Aldous (2013) concerning the distributions of the corresponding entropy processes and diversity processes for types.
We provide an original and general sufficient criterion ensuring the exponential contraction of Feynman-Kac semi-groups of penalized processes. This criterion is applied to time-inhomogeneous one-dimensional diffusion processes conditioned not to hit 0 and to penalized birth and death processes evolving in a quenched random environment.
Following Csörgő, Szyszkowicz and Wang (Ann. Statist. {\bf 34}, (2006), 1013--1044) we consider a long range dependent linear sequence. We prove weak convergence of the uniform Vervaat and the uniform Vervaat error processes, extending their results to distributions with unbounded support and removing normality assumption.
Invariance principles are obtained for a Markov process on a half-line with continuous paths on the interior. The domains of attraction of the two different types of self-similar processes are investigated. Our approach is to establish convergence of excursion point processes, which is based on Itô's excursion theory and a recent result on convergence of excursion measures by Fitzsimmons and the present author.
The paper studies the rate of convergence of the weak Euler approximation for solutions to SDEs driven by Levy processes, with Hoelder-continuous coefficients. It investigates the dependence of the rate on the regularity of coefficients and driving processes. The equation considered has a non-degenerate main part driven by a spherically-symmetric stable process.
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