Étude quantitative des chaînes de Markov par perturbation de leur noyau.

Abstract : The development of the modelling of the random phenomena using Markov chains raises the problem of the control of convergence of the algorithms of simulation. The methods of simulations by ergodic Markov chains is based on the law of large numbers, which stipulates that for any initial distribution and any function f, the empirical average converges to the average of f, calculated with the unique invariant probability of the chain. It is then advisable, to determine a sufficient number of step of simulation in order to approximate, in a relatively precise way, the average of a f by its empirical average. Several works studied the speed of convergence of the chain towards its steady state. However even in steady state, the problem of the control remains, as we want to obtain a confidence interval for a fixed level. Two approaches exist to determine a sufficient number of steps. Either by bounding directly the probability of deviation between the empirical average and the average of f under the invariant probability, or by using the CLT. The starting point of this thesis was the inequality of Gillman and more particularly the method used: namely tools presented in the book of Kato on the theory of the perturbation of the linear operators.
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Pascal Lezaud. Étude quantitative des chaînes de Markov par perturbation de leur noyau.. Probabilités [math.PR]. Université Toulouse 3 Paul Sabatier, 1998. Français. ⟨tel-01084797⟩

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