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A. Guillin

Publications and source records attributed to A. Guillin.

5 recordsLinked to original sources
arXiv2007

Convergence of adaptive mixtures of importance sampling schemes

In the design of efficient simulation algorithms, one is often beset with a poor choice of proposal distributions. Although the performance of a given simulation kernel can clarify a posteriori how adequate this kernel is for the problem at hand, a permanent on-line modification of kernels causes concerns about the validity of the resulting algorithm. While the issue is most often intractable for MCMC algorithms, the equivalent version for importance sampling algorithms can be validated quite precisely. We derive sufficient convergence conditions for adaptive mixtures of population Monte Carlo algorithms and show that Rao--Blackwellized versions asymptotically achieve an optimum in terms of a Kullback divergence criterion, while more rudimentary versions do not benefit from repeated updating.

math.ST↗
arXiv2005

Moderate deviations for particle filtering

Consider the state space model (X_t,Y_t), where (X_t) is a Markov chain, and (Y_t) are the observations. In order to solve the so-called filtering problem, one has to compute L(X_t|Y_1,...,Y_t), the law of X_t given the observations (Y_1,...,Y_t). The particle filtering method gives an approximation of the law L(X_t|Y_1,...,Y_t) by an empirical measure \frac{1}{n}\sum_1^nδ_{x_{i,t}}. In this paper we establish the moderate deviation principle for the empirical mean \frac{1}{n}\sum_1^nψ(x_{i,t}) (centered and properly rescaled) when the number of particles grows to infinity, enhancing the central limit theorem. Several extensions and examples are also studied.

math.PR↗
arXiv2005

Examples of moderate deviation principle for diffusion processes

Taking into account some likeness of moderate deviations (MD) and central limit theorems (CLT), we develop an approach, which made a good showing in CLT, for MD analysis of a family $$ S^κ_t=\frac{1}{t^κ}\int_0^tH(X_s)ds, \ t\to\infty $$ for an ergodic diffusion process $X_t$ under $0.5<κ<1$ and appropriate $H$. We mean a decomposition with ``corrector'': $$ \frac{1}{t^κ}\int_0^tH(X_s)ds={\rm corrector}+\frac{1}{t^κ}\underbrace{M_t}_{\rm martingale}. $$ and show that, as in the CLT analysis, the corrector is negligible but in the MD scale, and the main contribution in the MD brings the family ``$ \frac{1}{t^κ}M_t, t\to\infty. $'' Starting from Bayer and Freidlin, \cite{BF}, and finishing by Wu's papers \cite{Wu1}-\cite{WuH}, in the MD study Laplace's transform dominates. In the paper, we replace the Laplace technique by one, admitting to give the conditions, providing the MD, in terms of ``drift-diffusion'' parameters and $H$. However, a verification of these conditions heavily depends on a specificity of a diffusion model. That is why the paper is named ``Examples ...''.

math.PR↗
arXiv2003

MDP for integral functionals of fast and slow processes with averaging

We establish large deviation principle (LDP) for the family of vector-valued random processes $(X^ε,Y^ε),ε\to 0$ defined as $$ X^ε_t=\frac{1}{ε^κ}\int_0^t H(ξ^ε_s,Y^ε_s)ds, dY^ε_t=F(ξ^ε_t,Y^ε_t)dt+ Dε^{1/2-κ}G(ξ^ε_t,Y^ε_t)dW_t,$$ where $W_t$ is Wiener process and $ξ^ε_t$ is fast ergodic diffusion. We show that, under $κ<{1/2}$ or less and Veretennikov-Khasminskii type condition for fast diffusion, the LDP holds with rate function of Freidlin-Wentzell's type.

math.PR↗