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

Parking on a Random Tree

Abstract Consider an infinite tree with random degrees, i.i.d. over the sites, with a prescribed probability distribution with generating function G(s). We consider the following variation of Rényi's parking problem, alternatively called blocking RSA (random sequential adsorption): at every vertex of the tree a particle (or "car") arrives with rate one. The particle sticks to the vertex whenever the vertex and all of its nearest neighbors are not occupied yet. We provide an explicit expression for the so-called parking constant in terms of the generating function. That is, the occupation probability, averaged over dynamics and the probability distribution of the random trees converges in the large-time limit to (1 − α 2 )/2 with 1 α xdx G(x) = 1

Limit theorems for von Mises statistics of a measure preserving transformation

For a measure preserving transformation $T$ of a probability space $(X,\mathcal F,\mu)$ we investigate almost sure and distributional convergence of random variables of the form $$x \to \frac{1}{C_n} \sum_{i_1<n,...,i_d<n} f(T^{i_1}x,...,T^{i_d}x),\, n=1,2,...,$$ where $f$ (called the \emph{kernel}) is a function from $X^d$ to $\R$ and $C_1, C_2,...$ are appropriate normalizing constants. We observe that the above random variables are well defined and belong to $L_r(\mu)$ provided that the kernel is chosen from the projective tensor product $$L_p(X_1,\mathcal F_1, \mu_1) \otimes_{\pi}...\otimes_{\pi} L_p(X_d,\mathcal F_d, \mu_d)\subset L_p(\mu^d)$$ with $p=d\,r,\, r\ \in [1, \infty).$ We establish a form of the individual ergodic theorem for such sequences. Next, we give a martingale approximation argument to derive a central limit theorem in the non-degenerate case (in the sense of the classical Hoeffding's decomposition). Furthermore, for $d=2$ and a wide class of canonical kernels $f$ we also show that the convergence holds in distribution towards a quadratic form $\sum_{m=1}^{\infty} \lambda_m\eta^2_m$ in independent standard Gaussian variables $\eta_1, \eta_2,...$. Our results on the distributional convergence use a $T$--\,invariant filtration as a prerequisite and are derived from uni- and multivariate martingale approximations

Macroevolution of the plant–hummingbird pollination system

ABSTRACTPlant–hummingbird interactions are considered a classic example of coevolution, a process in which mutually dependent species influence each other's evolution. Plants depend on hummingbirds for pollination, whereas hummingbirds rely on nectar for food. As a step towards understanding coevolution, this review focuses on the macroevolutionary consequences of plant–hummingbird interactions, a relatively underexplored area in the current literature. We synthesize prior studies, illustrating the origins and dynamics of hummingbird pollination across different angiosperm clades previously pollinated by insects (mostly bees), bats, and passerine birds. In some cases, the crown age of hummingbirds pre‐dates the plants they pollinate. In other cases, plant groups transitioned to hummingbird pollination early in the establishment of this bird group in the Americas, with the build‐up of both diversities coinciding temporally, and hence suggesting co‐diversification. Determining what triggers shifts to and away from hummingbird pollination remains a major open challenge. The impact of hummingbirds on plant diversification is complex, with many tropical plant lineages experiencing increased diversification after acquiring flowers that attract hummingbirds, and others experiencing no change or even a decrease in diversification rates. This mixed evidence suggests that other extrinsic or intrinsic factors, such as local climate and isolation, are important covariables driving the diversification of plants adapted to hummingbird pollination. To guide future studies, we discuss the mechanisms and contexts under which hummingbirds, as a clade and as individual species (e.g. traits, foraging behaviour, degree of specialization), could influence plant evolution. We conclude by commenting on how macroevolutionary signals of the mutualism could relate to coevolution, highlighting the unbalanced focus on the plant side of the interaction, and advocating for the use of species‐level interaction data in macroevolutionary studies

Langevin diffusions on the torus: estimation and applications

We introduce stochastic models for continuous-time evolution of angles and develop their estimation. We focus on studying Langevin diffusions with stationary distributions equal to well-known distributions from directional statistics, since such diffusions can be regarded as toroidal analogues of the Ornstein–Uhlenbeck process. Their likelihood function is a product of transition densities with no analytical expression, but that can be calculated by solving the Fokker–Planck equation numerically through adequate schemes. We propose three approximate likelihoods that are computationally tractable: (i) a likelihood based on the stationary distribution; (ii) toroidal adaptations of the Euler and Shoji–Ozaki pseudo-likelihoods; (iii) a likelihood based on a specific approximation to the transition density of the wrapped normal process. A simulation study compares, in dimensions one and two, the approximate transition densities to the exact ones, and investigates the empirical performance of the approximate likelihoods. Finally, two diffusions are used to model the evolution of the backbone angles of the protein G (PDB identifier 1GB1) during a molecular dynamics simulation. The software package sdetorus implements the estimation methods and applications presented in the paper

AVONET: morphological, ecological and geographical data for all birds

Functional traits offer a rich quantitative framework for developing and testing theories in evolutionary biology, ecology and ecosystem science. However, the potential of functional traits to drive theoretical advances and refine models of global change can only be fully realised when species‐level information is complete. Here we present the AVONET dataset containing comprehensive functional trait data for all birds, including six ecological variables, 11 continuous morphological traits, and information on range size and location. Raw morphological measurements are presented from 90,020 individuals of 11,009 extant bird species sampled from 181 countries. These data are also summarised as species averages in three taxonomic formats, allowing integration with a global phylogeny, geographical range maps, IUCN Red List data and the eBird citizen science database. The AVONET dataset provides the most detailed picture of continuous trait variation for any major radiation of organisms, offering a global template for testing hypotheses and exploring the evolutionary origins, structure and functioning of biodiversity