Chapters 4 and 5 of the present monograph deal comprehensively with limit theorems for transient Markov chains. In Chapter 5 we consider drifts decreasing more slowly than 1/x and prove limit theorems including weak and strong laws of large numbers, convergence to normal distribution, functional convergence to Brownian motion, and asymptotic behaviour of the renewal measure.

Chapter 7 is the most conceptual part of the book. Our purpose here is to describe, without superfluous details, a change of measure strategy which allows us to transform a recurrent chain into a transient one, and vice versa. It is motivated by the exponential change of measure technique which goes back to Cramer. In the context of large deviations in collective risk theory, this technique allows us to transform a negatively drifted random walk into one with positive drift. Doob’s h-transform is the most natural substitute for an exponential change of measure in the context of Lamperti’s problem, that is, in the context of Markov chains with asymptotically zero drift.

Such transformations connect naturally previous chapters on asymptotic behaviour of transient chains with subsequent chapters, which are devoted to recurrent chains. A very important, in comparison with the classical Doob’s h-transform, the novelty consists in the fact that we use weight functions which are not necessarily harmonic, they are only asymptotically harmonic at infinity. The main challenge is to identify such functions under various drift scenarios.

In this chapter we present two spatial dependent models: one based on defining a latent variable for each area, and the other by defining one latent variable for each pair of latent areas. We call the latter the latent edges model. We compare both models with a real data set. Extensions to spatio-temporal constructions are also considered.

In this chapter we define what a conjugate family is in a Bayesian analysis context and develop detailed examples of some cases; in particular, we review the beta and binomial case, the Pareto and inverse Pareto case, the gamma and gamma case and the gamma and Poisson case. We conclude by providing a list of conjugate models.

In this chapter we show how to define temporal dependent sequences using a moving average type of construction. We compare the performance of this construction with a Markov-process type. We finally extend the models to include seasonal and periodic dependencies.

In this chapter we start with some attempts to construct dependence sequences with order larger than one and present a general result to achieve an invariant distribution via a three-level hierarchical model. We finally present some results involving exponential families.

In this chapter we describe a general procedure to construct Markov sequences with invariant distributions. The procedure can be used with conjugate and non-conjugate models and with parametric and nonparametric distributions. We derive several examples in detail and finish with some applications in survival analysis.

In this chapter we introduce the concept of exchangeability and show how to construct exchangeable sequences; we present our first result of how to construct exchangeable sequences and maintain a desirable marginal distribution and provide detailed examples. We finish with an application of exchangeable constructions in a meta analysis. Bugs and R code are provided.

In this chapter we start by reviewing the different types of inference procedures: frequentist, Bayesian, parametric and non-parametric. We introduce notation by providing a list of the probability distributions that will be used later on, together with their first two moments. We review some results on conditional moments and carry out several examples. We review definitions of stochastic processes, stationary processes and Markov processes, and finish by introducing the most common discrete-time stochastic processes that show dependence in time and space.

In this chapter we conclude the book by presenting dependent models for random vectors and for stochastic processes. The types of dependence are exchangeable, Markov, moving average, spatial or a combination of the latter two.