Markov tail chains

The extremes of a univariate Markov chain with regulary varying stationary marginal distribution and asymptotically linear behavior are known to exhibit a multiplicative random walk structure called the tail chain. In this paper, we extend this fact to Markov chains with multivariate regularly varying marginal distribution in R^d. We analyze both the forward and the backward tail process and show that they mutually determine each other through a kind of adjoint relation. In a broader setting, it will be seen that even for non-Markovian underlying processes a Markovian forward tail chain always implies that the backward tail chain is Markovian as well. We analyze the resulting class of limiting processes in detail. Applications of the theory yield the asymptotic distribution of both the past and the future of univariate and multivariate stochastic difference equations conditioned on an extreme event

Statistics for Tail Processes of Markov Chains

At high levels, the asymptotic distribution of a stationary, regularly varying Markov chain is conveniently given by its tail process. The latter takes the form of a geometric random walk, the increment distribution depending on the sign of the process at the current state and on the flow of time, either forward or backward. Estimation of the tail process provides a nonparametric approach to analyze extreme values. A duality between the distributions of the forward and backward increments provides additional information that can be exploited in the construction of more efficient estimators. The large-sample distribution of such estimators is derived via empirical process theory for cluster functionals. Their finite-sample performance is evaluated via Monte Carlo simulations involving copula-based Markov models and solutions to stochastic recurrence equations. The estimators are applied to stock price data to study the absence or presence of symmetries in the succession of large gains and losses

At the Edge of Criticality: Markov Chains with Asymptotically Zero Drift

In Chapter 2 we introduce a classification of Markov chains with asymptotically zero drift, which relies on relations between first and second moments of jumps. We construct an abstract Lyapunov functions which looks similar to functions which characterise the behaviour of diffusions with similar drift and diffusion coefficient. Chapter 3 is devoted to the limiting behaviour of transient chains. Here we prove converges to $\Gamma$ and normal distribution which generalises papers by Lamperti, Kersting and Klebaner. We also determine the asymptotic behaviour of the cumulative renewal function. In Chapter 4 we introduce a general strategy of change of measure for Markov chains with asymptotically zero drift. This is the most important ingredient in our approach to recurrent chains. Chapter 5 is devoted to the study of the limiting behaviour of recurrent chains with the drift proportional to $1/x$. We derive asymptotics for a stationary measure and determine the tail behaviour of recurrence times. All these asymptotics are of power type. In Chapter 6 we show that if the drift is of order $x^{-\beta}$ then moments of all orders $k\le [1/\beta]+1$ are important for the behaviour of stationary distributions and pre-limiting tails. Here we obtain Weibull-like asymptotics. In Chapter 7 we apply our results to different processes, e.g. critical and near-critical branching processes, risk processes with reserve-dependent premium rate, random walks conditioned to stay positive and reflected random walks. In Chapter 8 we consider asymptotically homogeneous in space Markov chains for which we derive exponential tail asymptotics

The asymptotic tails of limit distributions of continuous time Markov chains

This paper investigates tail asymptotics of stationary distributions and quasi-stationary distributions of continuous-time Markov chains on a subset of the non-negative integers. A new identity for stationary measures is established. In particular, for continuous-time Markov chains with asymptotic power-law transition rates, tail asymptotics for stationary distributions are classified into three types by three easily computable parameters: (i) Conley-Maxwell-Poisson distributions (light-tailed), (ii) exponential-tailed distributions, and (iii) heavy-tailed distributions. Similar results are derived for quasi-stationary distributions. The approach to establish tail asymptotics is different from the classical semimartingale approach. We apply our results to biochemical reaction networks (modeled as continuous-time Markov chains), a general single-cell stochastic gene expression model, an extended class of branching processes, and stochastic population processes with bursty reproduction, none of which are birth-death processes

Tail behaviour of stationary distribution for Markov chains with asymptotically zero drift

We consider a Markov chain on $R^+$ with asymptotically zero drift and finite second moments of jumps which is positive recurrent. A power-like asymptotic behaviour of the invariant tail distribution is proven; such a heavy-tailed invariant measure happens even if the jumps of the chain are bounded. Our analysis is based on test functions technique and on construction of a harmonic function.Comment: 27 page