Rate of convergence for the Donsker theorem for rough paths The Donsker theorem says that a random walk, when conveniently rescaled, converges to a Brownian motion. A finer result of Friz and Victoir, that the Lévy area of the random walk goes also to the Lévy area of the Brownian motion. The question is…
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One of many advantages of the Stein’s method is its versatility. It can be applied to prove convergence to Gaussian random variables or processes as well as to Poisson random variables or point processes. In this new paper, we investigate this last aspect showing that some multi-points transformation of some point processes lead to a…
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A few years ago, browsing the notices of the AMS, I found an article by R. Ghrist and V. de Silva about topological algebra used for detecting coverage problem. Barely stated, the difficulty is as follows: Consider a set of sensors, each of which detecting intrusion in a region around itself; how can we be…
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The Stein method is a generic method to evaluate distance between probability measures. It is often applied to prove convergence towards Gaussian or Poisson distribution. Its principle is summarized by the following graph. To compare the probability measures (m) and (m_beta), one constructs a Markov process which is ergodic, whose stationary distribution is (m_beta) and…
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Edgeworth series refine the central limit theorem. Playing with Stein method and Malliavin calculus, we can retrieve Edgeworth series and precise the error bounds. http://hal.archives-ouvertes.fr/hal-00685272 The infinite dimension version is under preparation.
We consider an SIR epidemic model propagating on a Configuration Model network, where the degree distribution of the vertices is given and where the edges are randomly matched. The evolution of the epidemic is summed up into three measure-valued equations that describe the degrees of the susceptible individuals and the number of edges from an…
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Algebraic topology has been extensively and successfully used in image processing for a long time. In telecommunication networks, it can be used to verify that a zone in the plane is fully covered by a set of base stations with a few informations on their relative locations. In this article, we compute some of the…
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Determinantal and permanental processes are point processes with a correlation function given by a determinant or a permanent. Their atoms exhibit mutual attraction of repulsion. These processes are thus very far from the uncorrelated situation encountered in Poisson models. Quasi-invariance In the linked paper, we establish a quasi-invariance result : we show that if atoms…
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