We establish a functional limit theorem for the joint-law of occupations near and away from indifferent fixed points of interval maps, and of waits for the occupations away from these points, in the sense of strong distributional convergence. It is a functional and joint-distributional extension of Darling--Kac type limit theorem, of Lamperti type generalized arcsine laws for occupation times, and of Dynkin and Lamperti type generalized arcsine laws for waiting times, at the same time.
We derive new variants of the quantitative Borel--Cantelli lemma and apply them to analysis of statistical properties for some dynamical systems. We consider intermittent maps of $(0,1]$ which have absolutely continuous invariant probability measures. In particular, we prove that every sequence of intervals with left endpoints uniformly separated from zero is the strong Borel--Cantelli sequence with respect to such map and invariant measure.
We study dynamical systems arising as time-dependent compositions of Pomeau-Manneville-type intermittent maps. We establish central limit theorems for appropriately scaled and centered Birkhoff-like partial sums, with estimates on the rate of convergence. For maps chosen from a certain parameter range, but without additional assumptions on how the maps vary with time, we obtain a self-normalized CLT provided that the variances of the partial sums grow sufficiently fast. When the maps are chosen randomly according to a shift-invariant probability measure, we identify conditions under which the …
We study the behavior of the empirical distribution function of iterates of intermittent maps in the Hilbert space of square inegrable functions with respect to Lebesgue measure. In the long-range dependent case, we prove that the empirical distribution function, suitably normalized, converges to a degenerate stable process, and we give the corresponding almost sure result. We apply the results to the convergence of the Wasserstein distance between the empirical measure and the invariant measure. We also apply it to obtain the asymptotic distribution of the corresponding Cramér-von-Mises stati…
We prove an abstract result establishing that one can obtain the convergence of Rare Events Point Processes counting the number of orbital visits to a sequence of shrinking target sets from the convergence of corresponding point processes for some induced system and matching shadowing shrinking sets inside the base of the inducing scheme. We apply this result to prove a dichotomy for two classes of non-uniformly hyperbolic interval maps: Misiurewicz quadratic maps and doubly intermittent maps. The dichotomy holds in the sense that the shrinking target sets may accumulate in any individual poin…
We compute some dependence coefficients for the stationary Markov chain whose transition kernel is the Perron-Frobenius operator of an expanding map $T$ of $[0, 1]$ with a neutral fixed point. We use these coefficients to prove a central limit theorem for the partial sums of $f\circ T^i$, when $f$ belongs to a large class of unbounded functions from $[0, 1]$ to ${\mathbb R}$. We also prove other limit theorems and moment inequalities.
We study non-stationary stochastic processes arising from sequential dynamical systems built on maps with a neutral fixed points and prove the existence of Extreme Value Laws for such processes. We use an approach developed in \cite{FFV16}, where we generalised the theory of extreme values for non-stationary stochastic processes, mostly by weakening the uniform mixing condition that was previously used in this setting. The present work is an extension of our previous results for concatenations of uniformly expanding maps obtained in \cite{FFV16}.
We consider a large class of piecewise expanding maps T of [0,1] with a neutral fixed point, and their associated Markov chain Y_i whose transition kernel is the Perron-Frobenius operator of T with respect to the absolutely continuous invariant probability measure. We give a large class of unbounded functions f for which the partial sums of f\circ T^i satisfy both a central limit theorem and a bounded law of the iterated logarithm. For the same class, we prove that the partial sums of f(Y_i) satisfy a strong invariance principle. When the class is larger, so that the partial sums of f\circ T^i…
This paper is about statistical properties of quasistatic dynamical systems. These are a class of non-stationary systems that model situations where the dynamics change very slowly over time due to external influence. We focus on the case where the time-evolution is described by intermittent interval maps (Pomeau-Manneville maps) with time-dependent parameters. In a suitable range of parameters, we obtain a description of the statistical properties as a stochastic diffusion, by solving a well-posed martingale problem. The results extend those of a related recent study due to Dobbs and Stenlund…
We prove a deviation bound for the maximum of partial sums of functions of $α$-dependent sequences as defined in Dedecker, Gou{ë}zel and Merlev{è}de (2010). As a consequence, we extend the Rosenthal inequality of Rio (2000) for $α$-mixing sequences in the sense of Rosenblatt (1956) to the larger class of $α$-dependent sequences. Starting from the deviation inequality, we obtain upper bounds for large deviations and an H{ö}lderian invariance principle for the Donsker line. We illustrate our results through the example of intermittent maps of the interval, which are not $α$-mixing in the sense o…
We prove that the joint distribution of the occupation time ratios for ergodic transformations preserving an infinite measure converges to a multidimensional version of Lamperti's generalized arcsine distribution, in the sense of strong distributional convergence. Our results can be applied to interval maps and Markov chains. We adopt the double Laplace transform method, which has been utilized in the study of occupation times of diffusions on multiray. We also discuss the inverse problem.
In this paper, we discuss the laws of the iterated logarithm (LIL) for occupation times of Markov processes $Y$ in general metric measure space both near zero and near infinity under some minimal assumptions. We first establish LILs of (truncated) occupation times on balls $B(x,r)$ of radii $r$ up to an function $Φ(r)$, which is an iterated logarithm of mean exit time of $Y$, by showing that the function $Φ$ is optimal. Our first result on LILs of occupation times covers both near zero and near infinity regardless of transience and recurrence of the process. Our assumptions are truly local in …
In the setting of intermittent Pomeau-Manneville maps with time dependent parameters, we show a functional correlation bound widely useful for the analysis of the statistical properties of the model. We give two applications of this result, by showing that in a suitable range of parameters the bound implies the conditions of the normal approximation methods of Stein and Rio. For a single Pomeau-Manneville map belonging to this parameter range, both methods then yield a multivariate central limit theorem with a rate of convergence.
In this paper, we give estimates of the quadratic transportation cost in the conditional central limit theorem for a large class of dependent sequences. Applications to irreducible Markov chains, dynamical systems generated by intermittent maps and $τ$-mixing sequences are given.
We study an intermittent quasistatic dynamical system composed of nonuniformly hyperbolic Pomeau--Manneville maps with time-dependent parameters. We prove an ergodic theorem which shows almost sure convergence of time averages in a certain parameter range, and identify the unique physical family of measures. The theorem also shows convergence in probability in a larger parameter range. In the process, we establish other results that will be useful for further analysis of the statistical properties of the model.
We consider time-dependent dynamical systems arising as sequential compositions of self-maps of a probability space. We establish conditions under which the Birkhoff sums for multivariate observations, given a centering and a general normalizing sequence $b(N)$ of invertible square matrices, are approximated by a normal distribution with respect to a metric of regular test functions. Depending on the metric and the normalizing sequence $b(N)$, the conditions imply that the error in the approximation decays either at the rate $O(N^{-1/2})$ or the rate $O(N^{-1/2} \log N)$, under the additional …
