Philippe Soulier

Practical drift conditions for subgeometric rates of convergence

We present a new drift condition which implies rates of convergence to the stationary distribution of the iterates of a \psi-irreducible aperiodic and positive recurrent transition kernel. This condition, extending a condition introduced by Jarner and Roberts [Ann. Appl. Probab. 12 (2002) 224-247] for polynomial convergence rates, turns out to be very convenient to prove subgeometric rates of convergence. Several applications are presented including nonlinear autoregressive models, stochastic unit root models and multidimensional random walk Hastings-Metropolis algorithms.

The FEXP estimator for potentially non-stationary linear time series

We consider semiparametric fractional exponential (FEXP) estimators of the memory parameter d for a potentially non-stationary linear long-memory time series with additive polynomial trend. We use differencing to annihilate the polynomial trend, followed by tapering to handle the potential non-invertibility of the differenced series. We propose a method of pooling the tapered periodogram which leads to more efficient estimators of d than existing pooled, tapered estimators. We establish asymptotic normality of the tapered FEXP estimator in the Gaussian case with or without pooling. We establish asymptotic normality of the estimator in the linear case if pooling is used. Finally, we consider minimax rate-optimality and feasible nearly rate-optimal estimators in the Gaussian case.

Moment bounds and central limit theorem for functions of Gaussian vectors

In this note, bounds for moments of functions of Gaussian vectors are proved, generalizing earlier results by Taqqu (Z. Wahrscheinlichkeitstheorie verw. Gebite 40 (1977) 203) and Arcones (Ann. probab. 15 (4) (1994) 2243). These bounds are used to derive a Lindeberg-Levy central limit theorem for triangular arrays of functions of Gaussian vectors. Statistical applications for long range dependent processes are given.

On the existence of some ARCH(∞) processes

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.

Heavy tailed time series with extremal independence

We consider heavy tailed time series whose finite-dimensional distributions are extremally independent in the sense that extremely large values cannot be observed consecutively. This calls for methods beyond the classical multivariate extreme value theory which is convenient only for extremally dependent multivariate distributions. We use the Conditional Extreme Value approach to study the effect of an extreme value at time zero on the future of the time series. In formal terms, we study the limiting conditional distribution of future observations given an extreme value at time zero. To this purpose, we introduce conditional scaling functions and conditional scaling exponents. We compute these quantities for a variety of models, including Markov chains, exponential autoregressive models, stochastic volatility models with heavy tailed innovations or volatilities.

Conditions for the Propagation of Memory Parameter from Durations to Counts and Realized Volatility

We establish sufficient conditions on durations that are stationary with finite variance and memory parameter

The periodogram of an i.i.d. sequence

d \in [0,1/2)

Estimation of bivariate excess probabilities for elliptical models

to ensure that the corresponding counting process

Computable convergence rates for sub-geometric ergodic Markov chains

N(t)

Limit conditional distributions for bivariate vectors with polar representation

satisfies