Search results for "PROB"

Additive functionals and push forward measures under Veretennikov's flow

2014

16 pages; In this work, we will be interested in the push forward measure $(\vf_t)_*\gamma$, where $\vf_t$ is defined by the stochastic differential equation \begin{equation*} d\vf_t(x)=dW_t + \ba(\vf_t(x))dt, \quad \vf_0(x)=x\in\mbR^m, \end{equation*} and $\gamma$ is the standard Gaussian measure. We will prove the existence of density under the hypothesis that the divergence $\div(\ba)$ is not a function, but a signed measure belonging to a Kato class; the density will be expressed with help of the additive functional associated to $\div(\ba)$.

MODERATE DEVIATION PRINCIPLES FOR KERNEL ESTIMATOR OF INVARIANT DENSITY IN BIFURCATING MARKOV CHAINS MODELS

2021

Bitseki and Delmas (2021) have studied recently the central limit theorem for kernel estimator of invariant density in bifurcating Markov chains models. We complete their work by proving a moderate deviation principle for this estimator. Unlike the work of Bitseki and Gorgui (2021), it is interesting to see that the distinction of the two regimes disappears and that we are able to get moderate deviation principle for large values of the ergodic rate. It is also interesting and surprising to see that for moderate deviation principle, the ergodic rate begins to have an impact on the choice of the bandwidth for values smaller than in the context of central limit theorem studied by Bitseki and …

Quantiles de régression : applications à la construction de courbes de référence

2006

Characterization of stationary probability measures for Variable Length Markov Chains

2020

By introducing a key combinatorial structure for words produced by a Variable Length Markov Chain (VLMC), the longest internal suffix, precise characterizations of existence and uniqueness of a stationary probability measure for a VLMC chain are given. These characterizations turn into necessary and sufficient conditions for VLMC associated to a subclass of probabilised context trees: the shift-stable context trees. As a by-product, we prove that a VLMC chain whose stabilized context tree is again a context tree has at most one stationary probability measure.

Algorithmes Stochastiques

2010

National audience; Un algorithme stochastique est un outil d'optimisation particulièrement utile lorsque l'on observe les données "en ligne". Il permet, entre autres choses, d'estimer les paramètres de modèles statistiques avec des procédures généralement simples de mise à jour qui ne nécessitent pas la ré-estimation parfois très coûteuse en temps de calcul. Ces approches itératives possèdent de bonnes propriétés de convergence et atteignent même dans certains cas la vitesse optimale. Dans le vaste domaine d'applications, citons l'économie, la finance, la biologie, la physique mathématique, l'automatique, le traitement d'images jusqu'à la statistique non paramétrique. L'objet de cette sessi…

Some multivariate risk indicators ; estimation and application to optimal reserve allocation

2010

National audience; We consider a vectorial risk process ...

Algorithmes stochastiques et diffusions : l'étude par inégalités fonctionnelles

2010

National audience; Les algorithmes classiques que sont le recuit simulé ou l'algorithme de Robbins- Monro ont des équivalents en temps continu, qui sont des processus de diffusion non-homogènes.

CENTRAL LIMIT THEOREM FOR KERNEL ESTIMATOR OF INVARIANT DENSITY IN BIFURCATING MARKOV CHAINS MODELS

2021

Bifurcating Markov chains (BMC) are Markov chains indexed by a full binary tree representing the evolution of a trait along a population where each individual has two children. Motivated by the functional estimation of the density of the invariant probability measure which appears as the asymptotic distribution of the trait, we prove the consistence and the Gaussian fluctuations for a kernel estimator of this density based on late generations. In this setting, it is interesting to note that the distinction of the three regimes on the ergodic rate identified in a previous work (for fluctuations of average over large generations) disappears. This result is a first step to go beyond the thresh…

CENTRAL LIMIT THEOREM FOR BIFURCATING MARKOV CHAINS

2020

Bifurcating Markov chains (BMC) are Markov chains indexed by a full binary tree representing the evolution of a trait along a population where each individual has two children. We first provide a central limit theorem for general additive functionals of BMC, and prove the existence of three regimes. This corresponds to a competition between the reproducing rate (each individual has two children) and the ergodicity rate for the evolution of the trait. This is in contrast with the work of Guyon (2007), where the considered additive functionals are sums of martingale increments, and only one regime appears. Our first result can be seen as a discrete time version, but with general trait evoluti…

Risk indicators with several lines of business: comparison, asymptotic behavior and applications to optimal reserve allocation

2013

International audience; In a multi-dimensional risk model with dependent lines of business, we propose to allocate capital with respect to the minimization of some risk indicators. These indicators are sums of expected penalties due to the insolvency of a branch while the global reserve is either positive or negative. Explicit formulas in the case of two branches are obtained for several models independent exponential, correlated Pareto). The asymptotic behavior (as the initial capital goes to infinity) is studied. For higher dimension and several periods, no explicit expression is available. Using a stochastic algorithm, we get estimations of the allocation, compare the different allocatio…