Large deviations for kernel density estimators and study for random decrement estimator

Overview & Synopsis

This thesis is devoted to the studies of two themes : large deviations of the kernel density estmator for stationary stochastic processes and random decrement estimator (RDE) for stationary gaussian processes. The first theme is the main part of this thesis and contains four chapters. In chapter 1, we establish the w*-LDP (large deviation principle) of $f_n^*$ and a concentration inequality in the i.i.d. case. In chapter 2, we prove the exponential convergence of $f_n^*$ dans $L^1(R^d)$ and a concentration inequality for the $\phi$-mixing processes, by using a transportation inequality of Rio. Chapter 3 and chapter 4 are the core of this thesis. For the first time in the dependent case, we establish (i) the LDP of $f_n^*$ for the weak topology $\sigma(L^1, L^{\infty})$; (ii) the weak*-LDP of $f_n^*$ in $L^1$ for the strong topology $\vert\cdot\vert_1$ ; (iii) large deviation estimations for $D_n^*=\vert f_n^*(x)-f(x) \vert_1$ and (iv) asymptotic optimality of $f_n^*$ in the sense of Bahadur. These results are established in chapter 3 for uniformly ergodic Markov processes; and for uniformly integrable reversible Markov processes in the chapter 4. The last chapter is devoted to the second theme. We prove the law of large number and central limit theorem for RDE in discret time case and give the explicit bias of the RDE in continuous time case.