Large Deviations Of Kernel Density Estimator For Stochastic Processes

This thesis is devoted to the study of large deviations for kernel density estimator for certain stochastic processes, especially in the dependent case. It generalizes for the first time the large deviation estimations from independent identical distributed(i.i.d. in short) case to dependent case.Given a stationary sample (X_k)k≥0 valued in R^d with the marginal law P(X₀ ∈ dx) = f(x)dx where the density f is unknown, the kernel density estimator of f is defined as usually as:where K is some given probability density function on R^d, and the bandwidth h = h_n, depending on n, satisfieIn the first part, the case of i.i.d. is studied, the large deviation principle (LDP in short) in the weak topology σ(L¹,L^∞) and the weak*-LDP in the strong L¹-topology for f_n^* are established. They are generalizations of the LDP of ||f_n^* - f||₁ established by Louani (2000). Furthermore, using the T₁ transportation inequality for product measures, we obtain a concentration inequality which is much finer than that of Devroye (1983).In the following parts, the dependent case is studied. Firstly, for the φ-mixing processes, as the LDP of empirical measures doesn't hold for general φ-mixing processes, we could not hope to obtain the LDP for /*. However under a quite general condition (the summability of the φ-mixing coefficients), we establish the exponential convergence of ||f_n^* - f||₁ which gives a first step to prove the LDP for certain φ-mixing processes. Secondly, Markov processes are studies. For the uniformly ergodic Markov processes, which is a classic framwork established by Deuschel-Stroock for studying the LDP for empirical measures, the LDP of f_n^* in the weak topology and the uniform weak*-LDP in the strong topology || · ||₁ for f_n^* are established. Furthermore, we prove that f_n^* is asymptotic optimal in the sense of Bahadur. These results are obtained for the first time in the dependent case. For this generalization from the i.i.d. case to dependent case, we have to overcome numerous new technical difficulties and to apply various tools: such as Harnack type inequality, Cramer type deviation inequality, uniformly integrable operators, perturbation theory of linear operators, Bishop-Phelps theorem, etc (those tools are used for the first time in this question). Finally, all the results in the uniformly ergodic case can be generalized to reversible Markov processes satisfying the uniform integrability condition.