In this paper, we investigate a nonparametric approach to provide a recursive estimatorof the transition density of a piecewise-deterministic Markov process, from only oneobservation of the path within a long time. In this framework, we do not observe a Markovchain with transition kernel of interest. Fortunately, one may write the transitiondensity of interest as the ratio of the invariant distributions of two embedded chains ofthe process. Our method consists in estimating these invariant measures. We state a resultof consistency and a central limit theorem under some general assumptions about the mainfeatures of the process. A simulation study illustrates the well asymptotic behavior ofour estimator.

An asymptotic model for the extreme behavior of certain Markov chains is the ‘tail chain’. Generally taking the form of a multiplicative random walk, it is useful in deriving extremal characteristics, such as point process limits. We place this model in a more general context, formulated in terms of extreme value theory for transition kernels, and extend it by formalizing the distinction between extreme and nonextreme states. We make the link between the update function and transition kernel forms considered in previous work, and we show that the tail chain model leads to a multivariate regular variation property of the finite-dimensional distributions under assumptions on the marginal tails alone.

We consider the first passage percolation problem on the random graph with vertex set N x {0, 1}, edges joining vertices at a Euclidean distance equal to unity, and independent exponential edge weights. We provide a central limit theorem for the first passage times ln between the vertices (0, 0) and (n, 0), thus extending earlier results about the almost-sure convergence of ln / n as n → ∞. We use generating function techniques to compute the n-step transition kernels of a closely related Markov chain which can be used to explicitly calculate the asymptotic variance in the central limit theorem.