Let N1 be a fixed integer and (C1,…,CN,Q) a random element of M(d×d,R)N×Rd. We consider solutions of multivariate smoothing transforms, i.e. random variables R satisfying R=d∑i=1NCiRi+Q where =d denotes equality in distribution, and R,R1,…,RN are independent identically distributed...

This paper is aimed at sharpen a weak invariance principle for stationary sequencesin Doukhan & Louhichi (1999). Our assumption is both beyond mixing and the causal-weak dependence in Dedecker and Doukhan (2003); those authors obtained a sharpresult which improves on an optimal one in Doukhan et...

We introduce the notion of continuously invertible volatility models that relies on some Lyapunov condition and some regularity condition. We show that it is almost equivalent to the volatilities forecasting efficiency of the parametric inference approach based on the Stochastic Recurrence...

We prove the existence of a weakly dependent strictly stationary solution of the equation Xt=F(Xt-1,Xt-2,Xt-3,...;[xi]t) called a chain with infinite memory. Here the innovations [xi]t constitute an independent and identically distributed sequence of random variables. The function F takes values...

This paper provides a probabilistic and statistical comparison of the log-GARCH and EGARCH models, which both rely on multiplicative volatility dynamics without positivity constraints. We compare the main probabilistic properties (strict stationarity, existence of moments, tails) of the EGARCH...

This paper provides a probabilistic and statistical comparison of the log-GARCH and EGARCH models, which both rely on multiplicative volatility dynamics without positivity constraints. We compare the main probabilistic properties (strict stationarity, existence of moments, tails) of the EGARCH...

This paper studies the probabilistic properties and the estimation of the asymmetric log-GARCH($p,q$) model. In this model, the log-volatility is written as a linear function of past values of the log-squared observations, with coefficients depending on the sign of the observations, and past...

We introduce the notion of continuous invertibility on a compact set for volatility models driven by a Stochastic Recurrence Equation (SRE). We prove the strong consistency of the Quasi Maximum Likelihood Estimator (QMLE) when the optimization procedure is done on a continuously invertible...