Periodogram ordinates of a Gaussian white-noise computed at Fourier frequencies are well known to form an i.i.d. sequence. This is no longer true in the non-Gaussian case. In this paper, we develop a full theory for weighted sums of non-linear functionals of the periodogram of an i.i.d....

This paper deals with a general class of observation-driven time series models with a special focus on time series of counts. We provide conditions under which there exist strict-sense stationary and ergodic versions of such processes. The consistency of the maximum likelihood estimators is then...

This paper discusses quantitative bounds on the convergence rates of Markov chains, under conditions implying polynomial convergence rates. This paper extends an earlier work by Roberts and Tweedie (Stochastic Process. Appl. 80(2) (1999) 211), which provides quantitative bounds for the total...

We establish a simple variance inequality for U-statistics whose underlying sequence of random variables is an ergodic Markov Chain. The constants in this inequality are explicit and depend on computable bounds on the mixing rate of the Markov Chain. We apply this result to derive the strong law...

The forgetting of the initial distribution for discrete Hidden Markov Models (HMM) is addressed: a new set of conditions is proposed, to establish the forgetting property of the filter, at a polynomial and geometric rate. Both a pathwise-type convergence of the total variation distance of the...

Let F be a distribution function in the maximal domain of attraction of the Gumbel distribution such that −log(1−F(x))=x1/θL(x) for a positive real number θ, called the Weibull tail index, and a slowly varying function L. It is well known that the estimators of θ have a very slow rate of...

This paper describes the limiting behaviour of tail empirical processes associated with long memory stochastic volatility models. We show that such a process has dichotomous behaviour, according to an interplay between the Hurst parameter and the tail index. On the other hand, the tail empirical...

A new sufficient condition for the existence of a stationary causal solution of an equation is provided. This condition allows us to consider coefficients with power-law decay, so that it can be applied to the so-called FIGARCH processes, whose existence is thus proved.