Regularity of the Characteristic Function of Additive Functionals for Iterated Function Systems. Statistical Applications

#### D. Ferre, D. Guibourg

2013, v.19, Issue 2, 299-342

ABSTRACT

Let $(\x,d)$ be a complete metric space, let $\v$ be a measurable space, and let $(X_n)_{n\in\n}$ be an iterated function system (IFS) of i.i.d. Lipschitz maps, namely $(X_n)_{n\in\n}$ is a sequence of $\x$-valued random variables (r.v.) recursively defined by $X_n = F(\vartheta_n,X_{n-1})$, where the sequence $(\vartheta_n)_{n\geq 1}$ of $\v$-valued i.i.d.r.v. is independent from the initial r.v. $X_0$, and where $F: \v\times \x\r \x$ is a measurable function such that for all $\vartheta\in \v$, $F(\vartheta,\cdot)$ has a finite Lipschitz constant $L_\vartheta$ on $(\x,d)$. Given $\xi: \v \times \x \rightarrow \r^N$ a weighted-Lipschitz functional, let $S_n =\sum_{k=1}^n\xi(\vartheta_k,X_{k-1})$ be the associated additive functional. Our main results concern the regularity of the characteristic function of $S_n$. The assumptions are expressed in term of contractive/moment conditions involving the r.v. $L_{\vartheta_1}$, $d(x_0,F_{\vartheta1}x_0)$ (for some fixed $x_0\in\x$) and the r.v. defined as the weighted tail and Lipschitz coefficient of $\xi(\vartheta_1,\cdot)$. When applied to classical models, these conditions reduce to (almost) expected moment conditions (in comparison with the i.i.d. case). Our approach appeals to the spectral analysis of Fourier operators. From this study, most of the classical limit theorems based on standard Fourier techniques can be deduced as in the i.i.d. case. In addition, if both the definition and the law of the IFS, as well as the functional $\xi$, depend on some parameters, then we obtain a precise control on constants involved in certain limit theorems, as for instance the Berry - Esseen bound. As an illustration, we prove the expected rate of convergence in the asymptotic normality for general $M$-estimators associated to IFS of i.i.d. Lipschitz maps. All our examples and applications show the interest of considering additive functionals of $(\vartheta_k,X_{k-1})$.

Keywords: Markov chains,Iterative model,spectral method,$M$-estimator