publications

2023

1. Stat Papers

   Learning CHARME models with neural networks

   José G. Gómez-García , Jalal Fadili , and Christophe Chesneau

   Statistical Papers, 2023

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   In this paper, we consider a model called CHARME (Conditional Heteroscedastic Autoregressive Mixture of Experts), a class of generalized mixture of nonlinear (non)parametric AR-ARCH time series. The main objective of this paper is to learn the autoregressive and volatility functions of this model with neural networks (NN). This approach is justified thanks to the universal approximation capacity of neural networks. On the other hand, in order to build the learning theory, it is necessary first to prove the ergodicity of the CHARME model. We therefore show in a general nonparametric framework that under certain Lipschitz-type conditions on the autoregressive and volatility functions, this model is stationary, ergodic and τ-weakly dependent. These conditions are much weaker than those in the existing literature. Moreover, this result forms the theoretical basis for deriving an asymptotic theory of the underlying parametric estimation, which we present for this model in a general parametric framework. Altogether, this allows to develop a learning theory for the NN-based autoregressive and volatility functions of the CHARME model, where strong consistency and asymptotic normality of the considered estimator of the NN weights and biases are guaranteed under weak conditions. Numerical experiments are reported to support our theoretical findings.

2021

1. Mathematics

   A Dependent Lindeberg Central Limit Theorem for Cluster Functionals on Stationary Random Fields

   José G. Gómez-García , and Christophe Chesneau

   Mathematics, 2021

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   In this paper, we provide a central limit theorem for the finite-dimensional marginal distributions of empirical processes (Zn(f))f∈F whose index set F is a family of cluster functionals valued on blocks of values of a stationary random field. The practicality and applicability of the result depend mainly on the usual Lindeberg condition and on a sequence Tn which summarizes the dependence between the blocks of the random field values. Finally, in application, we use the previous result in order to show the Gaussian asymptotic behavior of the proposed iso-extremogram estimator.

2020

1. Cyclostationarity

   On Extreme Values in Stationary Weakly Dependent Random Fields

   Paul Doukhan , and José G. Gómez-García

   In Cyclostationarity: Theory and Methods – IV , 2020

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   The existing literature on extremal types theorems for stationary random processes and fields is, until now, developed under either mixing or “Coordinatewise (Cw)-mixing” conditions. However, these mixing conditions are very restrictives and difficult to verify in general for many models. Due to these limitations, we extend the existing theory, concerning the asymptotic behaviour of the maximum of stationary random fields, to a weaker and simplest to verify dependence condition, called weak dependence, introduced by Doukhan and Louhichi [Stochastic Processes and their Applications 84 (1999): 313–342]. This stationary weakly dependent random fields family includes models such as Bernoulli shifts, chaotic Volterra and associated random fields, under reasonable addition conditions. We mention and check the weak dependence properties of some specific examples from this list, such as: linear, Markovian and LARCH(∞) fields. We show that, under suitable weak-dependence conditions, the maximum may be regarded as the maximum of an approximately independent sequence of sub-maxima, although there may be high local dependence leading to clustering of high values. These results on asymptotic max-independence allow us to prove an extremal types theorem and discuss domain of attraction criteria in this framework. Finally, a numerical experiment using a non-mixing weakly dependent random field is performed.

2018

1. Statistics

   Dependent Lindeberg central limit theorem for the fidis of empirical processes of cluster functionals

   José G. Gómez-García

   Statistics, 2018

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   Drees H. and Rootzén H. [Limit theorems for empirical processes of cluster functionals (EPCF). Ann Stat. 2010;38(4):2145–2186] have proven central limit theorems (CLTs) for EPCF built from β-mixing processes. However, this family of β-mixing processes is quite restrictive. We expand some of those results, for the finite-dimensional marginal distributions (fidis), to a more general dependent processes family, known as weakly dependent processes in the sense of Doukhan P. and Louhichi S. [A new weak dependence condition and applications to moment inequalities. Stoch. Proc. Appl. 1999;84:313–342]. In this context, the CLT for the fidis of EPCF is sufficient in some applications. For instance, we prove the convergence without mixing conditions of the extremogram estimator, including a small example with simulation of the extremogram of a weakly dependent random process but nonmixing, in order to confirm the efficacy of our result.

2017

1. Limit theorems for functionals of clusters of extremes and applications

   José G. Gómez-Garcı́a

   Cergy Paris Université/AGM , Nov 2017

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   This thesis deals mainly with limit theorems for empirical processes of extreme cluster functionals of weakly dependent random fields and sequences. Limit theorems for empirical processes of extreme cluster functionals of stationnary time series are given by Drees & Rootzén [2010] under absolute regularity (or "β-mixing") conditions. However, these dependence conditions of mixing type are very restrictive: on the one hand, they are best suited for models in finance and history, and on the other hand, they are difficult to verify. Generally, for other models common in applications, the mixing conditions are not satisfied. In contrast, weak dependence conditions, as defined by Doukhan & Louhichi [1999] and Dedecker & Prieur [2004a], are dependence conditions which generalises the notions of mixing and association. These are easier to verify and applicable to a wide list of models. More precisely, under weak conditions, all the causal or non-causal processes are weakly dependent: Gaussian, associated, linear, ARCH(∞), bilinear and Volterra processes are some included in this list. Under these conveniences, we expand some of the limit theorems of Drees & Rootzén [2010] to weakly dependent processes. These latter results are used in order to show the convergence in distribution of the extremogram estimator of Davis & Mikosch [2009] and the functional estimator of the extremal index introduced by Drees [2011] under weak dependence. We prove an extreme value theorem for weakly dependent stationary random fields and we propose, under the same conditions, a domain of attraction criteria of a law of extremes. The document ends with limit theorems for the empirical process of extreme cluster functionals of stationary weakly dependent random fields, deriving also the convergence in distribution of the estimator of an extremogram for stationary weakly dependent space-time processes.

2012

1. Una Teoría Matemática de Microlentes Estocásticos: Imágenes Aleatorias, Cortes Aleatorios y la Fórmula de Kac-Rice

   José G. Gómez-Garcı́a

   Central University of Venezuela/Mathematics School , Nov 2012