We give necessary conditions for geometric and polynomial convergence rates of randomwalk- type Markov chains to stationarity in terms of existence of exponential and polynomial moments of the invariant distribution and the Markov transition kernel. These results complement the use of Foster-Lyapunov drift conditions for establishing geometric and polynomial ergodicity. For polynomially ergodic Markov chains, the results allow us to derive exact rates of convergence and exact relations between the moments of the invariant distribution and the Markov transition kernel. In an application to Markov chain Monte Carlo we derive tight rates of convergence for symmetric random walk Metropolis.

A new nonparametric regression technique is proposed which involves the extension of local polynomial fitting to the empirical likelihood context, where the distribution of the stochastic error is not fully specified. The aim of this extension is to reduce the possible modelling bias of parametric likelihood and to allow one to use the auxiliary information about the stochastic error in the local polynomial fitting. The asymptotic bias and variance, consistency and asymptotic distribution of the proposed estimators are established. The proposed estimators are shown to inherit the main advantage of the local polynomial estimator based on the parametric likelihood over the Nadaraya-Watson kernel estimator near the boundaries. Moreover, the proposed estimators can be more flexible and efficient than the parametric likelihood based local polynomial estimator when the distribution of the stochastic error is misspecified. The new method is illustrated with applications to some simulated and real data sets.

We establish a bijection between exponentially convex functions and entire positive definite functions, and extend Lo$#x00E8;ve´s construction of stochastic processes associated with them. As an application, we derive parametric covariance models for locally stationary random fields.

The asymptotic normality of a class of estimators for extreme quantiles is established under mild structural conditions on the observed stationary β-mixing time series. Consistent estimators of the asymptotic variance are introduced, which render possible the construction of asymptotic confidence intervals for the extreme quantiles. Moreover, it is shown that many well-known time series models satisfy our conditions. The theory is then applied to a time series of stock index returns. Finally, the finite-sample behaviour of the proposed confidence intervals is examined in a simulation study. It turns out that for most time series models under consideration the actual coverage probability is pretty close to the nominal level if the sample fraction used for estimation is chosen appropriately.

We address the question of the optimal allocation of electricity production resources among several production technologies. We consider the effect of competition among different power generators on the equilibrium of the real-time electricity market. For this equilibrium optimal strategies of electricity producers are obtained, yielding for each technology the production capacity to be installed.

We study the superposition process of a class of independent renewal processes with longrange dependence. It is known that under two different scalings in time and space either fractional Brownian motion or a stable Lévy process may arise in the rescaling asymptotic limit. It is shown here that in a third, intermediate scaling regime a new limit process appears, which is neither Gaussian nor stable. The new limit process is characterized by its cumulant generating function and some of its properties are discussed.

We consider directed polymers in a random environment. Under some mild assumptions on the environment, we prove equivalence between the decay rate of the partition function and some natural localization properties of the path; some quantitative estimates of the decay of the partition function in one or two dimensions, or at sufficiently low temperature; and the existence of quenched free energy. In particular, we generalize to general environments the results recently obtained by Carmona and Hu for a Gaussian environment. Our approach is based on martingale decomposition and martingale analysis. It leads to a natural, asymptotic relation between the partition function, and the probability that two polymers in the same environment, but otherwise independent, end up at the same point.

A discrete-time approximation scheme called local linearization of the Langevin diffusion on Rk is considered, with emphasis on the ergodic properties of the approximation considered as a discrete-time Markov chain. We will derive criteria for the scheme to be geometrically ergodic, and illustrate the use of these criteria by means of examples. Furthermore, we discuss the scheme in relation to other schemes and the use of such discretization schemes as proposals in a Metropolis-Hastings algorithm.