Empirical central limit theorems for ergodic automorphisms of the torus

Let T be an ergodic automorphism of the d-dimensional torus T^d, and f be a continuous function from T^d to R^l. On the probability space T^d equipped with the Lebesgue-Haar measure, we prove the weak convergence of the sequential empirical process of the sequence (f o T^i)_{i \geq 1} under some mild conditions on the modulus of continuity of f. The proofs are based on new limit theorems and new inequalities for non-adapted sequences, and on new estimates of the conditional expectations of f with respect to a natural filtration.Comment: 32 page

Some unbounded functions of intermittent maps for which the central limit theorem holds

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.Comment: 16 page

An Empirical Process Central Limit Theorem for Multidimensional Dependent Data

Let $(U_n(t))_{t\in\R^d}$ be the empirical process associated to an $\R^d$-valued stationary process $(X_i)_{i\ge 0}$. We give general conditions, which only involve processes $(f(X_i))_{i\ge 0}$ for a restricted class of functions $f$, under which weak convergence of $(U_n(t))_{t\in\R^d}$ can be proved. This is particularly useful when dealing with data arising from dynamical systems or functional of Markov chains. This result improves those of [DDV09] and [DD11], where the technique was first introduced, and provides new applications.Comment: to appear in Journal of Theoretical Probabilit

Berry-Esseen type bounds for the Left Random Walk on GL d (R) under polynomial moment conditions

Let $A_n= \varepsilon_n \cdots \varepsilon_1$, where $(\varepsilon_n)_{n \geq 1}$ is a sequence of independent random matrices taking values in $GL_d(\mathbb R)$, $d \geq 2$, with common distribution $\mu$. In this paper, under standard assumptions on $\mu$ (strong irreducibility and proximality), we prove Berry-Esseen type theorems for $\log ( \Vert A_n \Vert)$ when $\mu$ has a polynomial moment. More precisely, we get the rate $((\log n) / n)^{q/2-1}$ when $\mu$ has a moment of order $q \in ]2,3]$ and the rate $1/ \sqrt{n}$ when $\mu$ has a moment of order $4$, which significantly improves earlier results in this setting

Rates in almost sure invariance principle for nonuniformly hyperbolic maps

We prove the Almost Sure Invariance Principle (ASIP) with close to optimal error rates for nonuniformly hyperbolic maps. We do not assume exponential contraction along stable leaves, therefore our result covers in particular slowly mixing invertible dynamical systems as Bunimovich flowers, billiards with flat points as in Chernov and Zhang (2005) and Wojtkowski' (1990) system of two falling balls. For these examples, the ASIP is a new result, not covered by prior works for various reasons, notably because in absence of exponential contraction along stable leaves, it is challenging to employ the so-called Sinai's trick (Sinai 1972, Bowen 1975) of reducing a nonuniformly hyperbolic system to a nonuniformly expanding one. Our strategy follows our previous papers on the ASIP for nonuniformly expanding maps, where we build a semiconjugacy to a specific renewal Markov shift and adapt the argument of Berkes, Liu and Wu (2014). The main difference is that now the Markov shift is two-sided, the observables depend on the full trajectory, both the future and the past

Targeting tillage intensity in Michigan soybean systems: On-farm observations and multivariate modeling of grower decision-making with implications for yield and soil carbon

Soybean growers must balance multiple, sometimes competing, economic and environmental objectives when deciding what level of tillage intensity is appropriate for a given field. Research has shown that decreasing soil disturbance can reduce the cost of soybean production, but the effects of conservation tillage on the soil environment and soybean performance are elusively site-specific, making precise tillage recommendations difficult. Moreover, grower decision-making regarding tillage intensity is a socio-psychological process whereby an individual’s attitude, beliefs, and social status augment their capacity for rational utility maximization. This study aims to illuminate how soybean growers in the State of Michigan select tillage technologies, and the effect of conservation tillage on key measures of agroecological performance in the field. Building on existing work in behavioral economics, human ecology, agricultural engineering, agronomy and soil science, it asks: What factors influence Michigan soybean growers’ selection of tillage technologies, and how do selected tillage technologies interact with variation in management history and the extant biophysical environment to affect soybean yield and soil organic carbon as integrated measures of agroecological function? In the context of three local ‘learning communities’ facilitated by Extension, thirty-five Michigan soybean growers were surveyed and on-farm observations of crop, soil and environmental variables collected from one hundred and thirty-three of their commercial soybean fields over a period of two growing seasons. Analysis of this large biophysical and social data set using a combination of behavioral, mixed and structural modeling demonstrated that the effects of a particular tillage system on soybean yield and soil carbon are indeed site-specific at the sub-field level, and that grower selection of tillage technologies is influenced by both economic and social factors. These results indicate that adapting tillage technologies to the environmental and social context in which they will be applied is critical to realizing the full potential of conservation tillage and its positive contributions to agricultural sustainability. On this basis, it is recommended that outreach promoting conservation tillage in Michigan target resource limited, experienced soybean growers with loose social network ties, and farms growing soybeans on poor quality soils in warmers areas of the State

Homogeneous variational problems: a minicourse

A Finsler geometry may be understood as a homogeneous variational problem, where the Finsler function is the Lagrangian. The extremals in Finsler geometry are curves, but in more general variational problems we might consider extremal submanifolds of dimension $m$. In this minicourse we discuss these problems from a geometric point of view.Comment: This paper is a written-up version of the major part of a minicourse given at the sixth Bilateral Workshop on Differential Geometry and its Applications, held in Ostrava in May 201