Sobolev-Type Classes of Functions with Values in a Metric Space. II

We prove equivalence of the definitions by the author and by Korevaar and Schoen of the Sobolev classes of mappings of a domain of an arithmetic n-dimensional space to a metric space.     

Empirical Polygon Simulation and Central Limit Theorems for the Homogenous Poisson Line Process

For the Poisson line process the empirical polygon is defined thanks to ergodicity and laws of large numbers for some characteristics, such as the number of edges, the perimeter, the area, etc... One also has, still thanks to the ergodicity of the Poisson line process, a canonical relation between this empirical polygon and the polygon containing a given point. The aim of this paper is to provide numerical simulations for both methods: in a previous paper (Paroux, Advances in Applied Probability, 30:640–656, 1998) one of the authors proved central limit theorems for some geometrical quantities associated with this empirical Poisson polygon, we provide numerical simulations for this phenomenon which generates billions of polygons at a small computational cost. We also give another direct simulation of the polygon containing the origin, which enables us to give further values for empirical moments of some geometrical quantities than the ones that were known or computed in the litterature. This work was partially supported by the PSMN at ENS-Lyon.     

Functional Central Limit Theorems for Triangular Arrays of Function-Indexed Processes under Uniformly Integrable Entropy Conditions

Functional central limit theorems for triangular arrays of rowwise independent stochastic processes are established by a method replacing tail probabilities by expectations throughout. The main tool is a maximal inequality based on a preliminary version proved by P. Gaenssler and Th. Schlumprecht. Its essential refinement used here is achieved by an additional inequality due to M. Ledoux and M. Talagrand. The entropy condition emerging in our theorems was introduced by K. S. Alexander, whose functional central limit theorem for so-calledmeasure-like processeswill be also regained. Applications concern, in particular, so-calledrandom measure processeswhich include function-indexed empirical processes and partial-sum processes (with random or fixed locations). In this context, we obtain generalizations of results due to K. S. Alexander, M. A. Arcones, P. Gaenssler, and K. Ziegler. Further examples include nonparametric regression and intensity estimation for spatial Poisson processes.     

Functional Central Limit Theorems for Supercritical Superprocesses

In this paper, we establish some functional central limit theorems for a large class of general supercritical superprocesses with spatially dependent branching mechanisms satisfying a second moment condition. In the particular case when the state \(E\) is a finite set and the underlying motion is an irreducible Markov chain on \(E\), our results are superprocess analogs of the functional central limit theorems of Janson (Stoch. Process. Appl. 110:177–245, 2004) for supercritical multitype branching processes. The results of this paper are refinements of the central limit theorems in Ren et al. (Stoch. Process. Appl. 125:428–457, 2015).     

Central Limit Theorems for Markov-Connected Random Variables

The paper provides some central limit theorems for triangular arrays of Markov-connected random variables. It is assumed the Markov chain to satisfy condition (D₁) which is an generalization of strong Doeblin's condition (D_o). One result represents a central limit theorem without the assumption of finite variances.     

Central Limit Theorems for Super Ornstein-Uhlenbeck Processes

Suppose that X={X _t :t≥0} is a supercritical super Ornstein-Uhlenbeck process, that is, a superprocess with an Ornstein-Uhlenbeck process on $\mathbb{R}^{d}$ corresponding to $L=\frac{1}{2}\sigma^{2}\Delta-b x\cdot\nabla$ as its underlying spatial motion and with branching mechanism ψ(λ)=?αλ+βλ ²+∫_(0,+∞)(e ^?λx ?1+λx)n(dx), where α=?ψ′(0+)>0, β≥0, and n is a measure on (0,∞) such that ∫_(0,+∞) x ² n(dx)<+∞. Let $\mathbb{P} _{\mu}$ be the law of X with initial measure μ. Then the process W _t =e ^?αt ∥X _t ∥ is a positive $\mathbb{P} _{\mu}$ -martingale. Therefore there is W _∞ such that W _t →W _∞, $\mathbb{P} _{\mu}$ -a.s. as t→∞. In this paper we establish some spatial central limit theorems for X. Let $\mathcal{P}$ denote the function class $$ \mathcal{P}:=\bigl\{f\in C\bigl(\mathbb{R}^d\bigr): \mbox{there exists } k\in\mathbb{N} \mbox{ such that }|f(x)|/\|x\|^k\to 0 \mbox{ as }\|x\|\to\infty \bigr\}. $$ For each $f\in\mathcal{P}$ we define an integer γ(f) in term of the spectral decomposition of f. In the small branching rate case α<2γ(f)b, we prove that there is constant $\sigma_{f}^{2}\in (0,\infty)$ such that, conditioned on no-extinction, $$\begin{aligned} \biggl(e^{-\alpha t}\|X_t\|, ~\frac{\langle f , X_t\rangle}{\sqrt{\|X_t\|}} \biggr) \stackrel{d}{\rightarrow}\bigl(W^*,~G_1(f)\bigr), \quad t\to\infty, \end{aligned}$$ where W ^? has the same distribution as W _∞ conditioned on no-extinction and $G_{1}(f)\sim \mathcal{N}(0,\sigma_{f}^{2})$ . Moreover, W ^? and G ₁(f) are independent. In the critical rate case α=2γ(f)b, we prove that there is constant $\rho_{f}^{2}\in (0,\infty)$ such that, conditioned on no-extinction, $$\begin{aligned} \biggl(e^{-\alpha t}\|X_t\|, ~\frac{\langle f , X_t\rangle}{t^{1/2}\sqrt{\|X_t\|}} \biggr) \stackrel{d}{\rightarrow}\bigl(W^*,~G_2(f)\bigr), \quad t\to\infty, \end{aligned}$$ where W ^? has the same distribution as W _∞ conditioned on no-extinction and $G_{2}(f)\sim \mathcal{N}(0, \rho_{f}^{2})$ . Moreover W ^? and G ₂(f) are independent. We also establish two central limit theorems in the large branching rate case α>2γ(f)b. Our central limit theorems in the small and critical branching rate cases sharpen the corresponding results in the recent preprint of Mi?o? in that our limit normal random variables are non-degenerate. Our central limit theorems in the large branching rate case have no counterparts in the recent preprint of Mi?o?. The main ideas for proving the central limit theorems are inspired by the arguments in K. Athreya’s 3 papers on central limit theorems for continuous time multi-type branching processes published in the late 1960’s and early 1970’s.     

Central Limit Theorems for Gromov Hyperbolic Groups

In this paper we study asymptotic properties of symmetric and nondegenerate random walks on transient hyperbolic groups. We prove a central limit theorem and a law of iterated logarithm for the drift of a random walk, extending previous results by S. Sawyer and T. Steger and of F. Ledrappier for certain CAT(−1)-groups. The proofs use a result by A. Ancona on the identification of the Martin boundary of a hyperbolic group with its Gromov boundary. We also give a new interpretation, in terms of Hilbert metrics, of the Green metric, first introduced by S. Brofferio and S. Blachère.     

Central Limit Theorems for Multiple Skorokhod Integrals

In this paper, we prove a central limit theorem for a sequence of multiple Skorokhod integrals using the techniques of Malliavin calculus. The convergence is stable, and the limit is a conditionally Gaussian random variable. Some applications to sequences of multiple stochastic integrals, and renormalized weighted Hermite variations of the fractional Brownian motion are discussed.     

To the Theory of Sobolev-Type Classes of Functions with Values in a Metric Space

We study some classes of functions with values in a complete metric space which can be considered as analogs of the Sobolev spaces W _p ¹ . Earlier the author considered the case of functions on a domain of ? ⁿ . Here we study the general case of mappings on an arbitrary Lipschitz manifold. We give necessary auxiliary facts, consider some examples, and describe some methods of construction of lower semicontinuous functionals on the classes W _p ¹ (M), where M is a Lipschitz manifold.     

Rates in the Empirical Central Limit Theorem for Stationary Weakly Dependent Random Fields

A weak dependence condition is derived as the natural generalization to random fields on notions developed in Doukhan and Louhichi (1999). Examples of such weakly dependent fields are defined. In the context of a weak dependence coefficient series with arithmetic or geometric decay, we give explicit bounds in Prohorov metric for the convergence in the empirical central limit theorem. For random fields indexed by &Zopf ^d , in the geometric decay case, rates have the form n ^−1/(8d+24) L(n), where L(n) is a power of log(n). This revised version was published online in June 2006 with corrections to the Cover Date.     

Central Limit Theorems for Supercritical Superprocesses with Immigration

In this paper, we establish a central limit theorem for a large class of general supercritical superprocesses with immigration with spatially dependent branching mechanisms satisfying a second moment condition. This central limit theorem extends and generalizes the results obtained by Ren et al. (Stoch Process Appl 125:428–457, 2015). We first give laws of large numbers for supercritical superprocesses with immigration since there are few convergence results on immigration superprocesses, then based on these results, we establish the central limit theorem.     

Central Limit Theorems for dependent variables. I

Summary This paper gives a flexible approach to proving the Central Limit Theorem (C.L.T.) for triangular arrays of dependent random variables (r.v.s) which satisfy a weak mixing condition called -mixing. Roughly speaking, an array of real r.v.s is said to be -mixing if linear combinations of its past and future are asymptotically independent. All the usual mixing conditions (such as strong mixing, absolute regularity, uniform mixing, -mixing and -mixing) are special cases of -mixing. Linear processes are shown to be -mixing under weak conditions. The main result makes no assumption of stationarity. A secondary result generalises a C.L.T. that Rosenblatt gave for strong mixing samples which are nearly second order stationary.     

Central Limit Theorems for Uniform Model Random Polygons

We show how a central limit theorem for Poisson model random polygons implies a central limit theorem for uniform model random polygons. To prove this implication, it suffices to show that in the two models, the variables in question have asymptotically the same expectation and variance. We use integral geometric expressions for these expectations and variances to reduce the desired estimates to the convergence $(1+\frac{\alpha}{n})^{n}\to e^{\alpha}$ as n????.     

Central and Functional Central Limit Theorems for a Class of Urn Models

We consider an approach based on tails to certain central limit and functional central limit theorems for a class of two color urn models. In particular, some of the results are derived from an associated Ornstein–Uhlenbeck process, and for another result we give an alternative proof based on martingale tails.      

Functional Non-Central and Central Limit Theorems for Bivariate Appell Polynomials

We consider quadratic forms in bivariate Appell polynomials involving strongly dependent time series. Both the spectral density of these time series and the Fourier transform of the kernel of the quadratic forms are regularly varying at the origin and hence may diverge, for example, like a power function. We obtain functional limit theorems for these quadratic forms by extending the recent results on the convergence of their finite-dimensional distributions. Some of these are functional central limit theorems where the limiting process is Brownian motion. Others are functional non-central limit theorems where the limiting processes are typically not Gaussian or, if they are Gaussian, then they are not Brownian motion.     

Central Limit Theorems for Asymptotically Negatively Associated Random Fields

Abstract The aim of this paper is to investigate the central limit theorems for asymptotically negatively dependent random fields under lower moment conditions or the Lindeberg condition. Results obtained improve a central limit theorem of Roussas [11] for negatively assiated fields and the main results of Su and Chi [18], and also include a central limit of theorem for weakly negatively associated random variables similar to that of Burton et al. [20]. Research supported by National Natural Science Foundation of China (No. 19701011)     

Central Limit Theorems for Random Permanents with Correlation Structure

Central limit theorems for permanents of random m×n matrices of iid columns with a common intercomponent correlation as n–m are derived. The results are obtained by introducing a Hoeffding-like orthogonal decomposition of a random permanent and deriving the variance formulae for a permanent with the homogeneous correlation structure.     

Almost Sure Central Limit Theorems for Heavily Trimmed Sums

We obtain an almost sure central limit theorem (ASCLT) for heavily trimmed sums. We also prove a function-typed ASCLT under the same conditions that assure measurable functions to satisfy the ASCLT for the partial sums of i.i.d, random variables with E X1 = 0, EX1^2 = 1.