Empirical Central Limit Theorem Research Articles

Distribution of levels in high-dimensional random landscapes

We prove empirical central limit theorems for the distribution of levels of various random fields defined on high-dimensional discrete structures as the dimension of the structure goes to ∞. The random fields considered include costs of assignments, weights of Hamiltonian cycles and spanning trees, energies of directed polymers, locations of particles in the branching random walk, as well as energies in the Sherrington–Kirkpatrick and Edwards–Anderson models. The distribution of levels in all models listed above is shown to be essentially the same as in a stationary Gaussian process with regularly varying nonsummable covariance function. This type of behavior is different from the Brownian bridge-type limit known for independent or stationary weakly dependent sequences of random variables.

Erratum to: An empirical central limit theorem for intermittent maps

There are two problems in the paper “An empirical central limit theorem for intermittent maps” published in Probability Theory and Related Fields (2010) 148:177-195. First, the definition of the coefficient β2(n) is in fact too restrictive. Secondly, there is a wrong argument in the proof of the main result, Theorem 2.1. In this erratum, we give the correct definition of the coefficient β2(n), as it was introduced in Dedecker and Prieur (2007), and we explain how to fix the proof of Theorem 2.1. The first paragraph is devoted to the definition of the coefficients. In the second paragraph, we give a slightly more general Rosenthal-type inequality than that given in Proposition 3.1, which will be used to fix the proof of Theorem 2.1. In the third paragraph, we explain the changes in the proof of Theorem 2.1.

An empirical Central Limit Theorem in [formula omitted] for stationary sequences

In this paper, we derive asymptotic results for the L 1 -Wasserstein distance between the distribution function and the corresponding empirical distribution function of a stationary sequence. Next, we give some applications to dynamical systems and causal linear processes. To prove our main result, we give a Central Limit Theorem for ergodic stationary sequences of random variables with values in L 1 . The conditions obtained are expressed in terms of projective-type conditions. The main tools are martingale approximations.

An empirical central limit theorem for intermittent maps

We prove an empirical central limit theorem for the distribution function of a stationary sequence, under a dependence condition involving only indicators of half line. We show that the result applies to the empirical distribution function of iterates of expanding maps with a neutral fixed point at zero as soon as the correlations are summable.

An empirical central limit theorem with applications to copulas under weak dependence

We state a multidimensional Functional Central Limit Theorem for weakly dependent random vectors. We apply this result to copulas. We get the weak convergence of the empirical copula process and of its smoothed version. The finite dimensional convergence of smoothed copula densities is also proved. A new definition and the theoretical analysis of conditional copulas and their empirical counterparts are provided.

Another look at the disjoint blocks bootstrap

We revisit the Disjoint Blocks Bootstrap and establish Empirical Central Limit Theorem under minimal conditions. This result is much more general than similar results for the Moving Blocks Bootstrap and nicely complements the results for independent variables. We perform a comprehensive numerical study comparing these two methods.

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.

An empirical central limit theorem for dependent sequences

We prove a central limit theorem for the d -dimensional distribution function of a class of stationary sequences. The conditions are expressed in terms of some coefficients which measure the dependence between a given σ -algebra and indicators of quadrants. These coefficients are weaker than the corresponding mixing coefficients, and can be computed in many situations. In particular, we show that they are well adapted to functions of mixing sequences, iterated random functions, and a class of dynamical systems.

Empirical central limit theorems for exchangeable random variables

This paper provides an empirical central limit theorem (CLT) for exchangeable random variables when the norming constants in the regular CLT are n h(n) with h( n) a slowly varying function. The situation with norming constants of the form nh( n) is also discussed.

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).

Can We Bootstrap Even if CLT Fails

It is shown that in some cases the bootstrap in probability holds under conditions weaker than necessary for the original empirical Central Limit Theorem.

An Introduction to Functional Central Limit Theorems for Dependent Stochastic Processes

This paper shows how the modern machinery for generating abstract empirical central limit theorems can be applied to arrays of dependent variables. It develops a bracketing approximation (closely related to results of Philipp and Massart) based on a moment inequality for sums of strong mixing arrays, in an effort to illustrate the sorts of difficulty that need to be overcome when adapting the empirical process theory for independent variables. Some suggestions for further development are offered. The paper is largely self-contained.

A uniform law of large numbers and empirical central limit theorem for limits of finite populations

Let U k be a finite poulation of size N k with elements a kj , 1≤j≤N k , all contained in a space U, and let (Y k1 , ..., Y k,Nk ) be one of the N k ! permutations of (a k1 , ... a k,Nk ) chosen with equal probabilities. The vector (Y k1 , ..., Y k,nk ), n k ≤N k , is an (ordered) simple random sample (SRS) from the population U k . Our main results are a uniform law of large numbers and an empirical central limit theorem for the array {Y kj : 1≤j≤n k }

A uniform CLT for uniformly bounded families of martingale differences

This paper develops a method to get empirical central limit theorems for martingale differences that are uniformly bounded. The main idea is to generalize to martingales some ideas of E. Gine and J. Zinn [Ann. Prob.12, 929–989 (1984)]. We consider two examples: An extension of a theorem of R. Dudley from i.i.d. to a certain type of Markov chain, and a uniform CLT for the “baker's transformation”.

Universal asymptotic normality for conditionally trimmed sums

The conditionally trimmed sums formed from an arbitrary i.i.d. sample are shown to satisfy both a probabilistic and empirical central limit theorem with a normal limit law. The specific method of trimming attempts to retain as many summands as possible and deletes only terms of sufficient magnitude. The behavior of the deleted terms is also studied for random variables which generate affinely stochastically compact partial sums.