Domain Of Normal Attraction Research Articles

Speed of convergence to equilibrium in Wasserstein metrics for Kac-like kinetic equations

This work deals with a class of one-dimensional measure-valued kinetic equations, which constitute extensions of the Kac caricature. It is known that if the initial datum belongs to the domain of normal attraction of an $\alpha$-stable law, the solution of the equation converges weakly to a suitable scale mixture of centered $\alpha$-stable laws. In this paper we present explicit exponential rates for the convergence to equilibrium in Kantorovich-Wasserstein distancesof order $p>\alpha$, under the natural assumption that the distancebetween the initial datum and the limit distribution is finite. For $\alpha=2$ this assumption reduces to the finiteness of the absolute moment of order $p$ of the initial datum. On the contrary, when $\alpha \alpha$. For this case, we provide sufficient conditions for the finiteness of the Kantorovich-Wasserstein distance.

Limit theory for a general class of GARCH models with just barely infinite variance

In this article, limit theory is established for a general class of generalized autoregressive conditional heteroskedasticity models given by ɛt = σtηt and σt = f (σt−1, σt−2,…, σt−p, ɛt−1, ɛt−2,…, ɛt−q), when {ɛt} is a process with just barely infinite variance, that is, {ɛt} is a process with infinite variance but in the domain of normal attraction. In particular, we show that under some regular conditions, converges weakly to a Gaussian process. Applications of the asymptotic results to statistical inference, such as unit root test and sample autocorrelation, are also investigated. The obtained result fills in a gap between the classical infinite variance and finite variance in the literature. Further, when applying our limiting result to Dickey–Fuller (DF) test in a unit root model with integrated generalized autoregressive conditional heteroskedasticity (IGARCH) errors, it just confirms the simulation result of Kourogenis and Pittis (2008) that the DF statistics with IGARCH errors converges in law to the standard DF distribution.

On the limit behavior of increments of sums of independent random variables from domains of attraction of asymmetric stable distributions

The asymptotic behavior of increments of sums of independent identically distributed random variables with incremental length (logn)p is considered. The laws describing increments of such length are intermediate between the Csogő-Revesz law (for large incremental lengths) and the Erdo-Renyi law (for small incremental lengths). A new result for random variables from the domain of normal attraction of asymmetric stable laws with parameter α e (1, 2) is obtained.

Unit root tests and dramatic shifts with infinite variance processes

A model which explains data that is subject to sudden structural changes of unspecified nature is presented. The structural shifts are generated by a random walk component whose innovations belong to the normal domain of attraction of a symmetric stable law. To test the model against the stationarity case, several non-parametric, and regression-based statistics are studied. The non-parametric tests are a generalization of the variance ratio test to innovations with heavy-tailed distributions. The tests are consistent and shown to have good finite sample size and power properties and are applied to a set of economic variables.

A Functional LIL for d-Dimensional Stable Processes; Invariance for Lévy- and Other Weakly Convergent Processes

We establish a functional LIL for the maximal process M(t) :=sup 0≤s≤t ‖X(s)‖ of an ℝ d -valued α-stable Levy process X, provided X(1) has density bounded away from zero over some neighborhood of the origin. We also provide a broad invariance result governing a class independent-increment processes related to the domain of attraction of X(1). This breadth is particularly notable for two types of processes captured: First, it not only describes any partial sum process built from iid summands in the domain of normal attraction of X(1), but also addresses those with arbitrary iid summands in the full domain of attraction (here we give a technical condition necessary and sufficient for the partial sum process to share the exact LIL we prove for X). Second, it reveals that any Levy process L such that L(1) satisfies the technical condition just mentioned will also share the LIL of X.

On Convergence in Law of Maxima of Independent Identically Distributed Random Variables with Random Coefficients

This paper provides limit theorems for maxima of independent identically distributed random variables which belong to the domain of normal attraction of a max-stable variable and have random coefficients with the property of asymptotic negligibility or its analogues. The convergence of random step-functions determined by those maxima is studied in the Skorokhod space. The limit distributions are indicated.

On estimation of the spectral measure of certain nonnormal operator stable laws

Abstract For data belonging to the domain of normal attraction of nonnormal operator stable laws we present a strongly consistent estimate of the s pectral measure. The cases of a known or unknown exponent are considered.

TESTING FOR ZERO AUTOCORRELATION WHEN THE INNOVATIONS BELONG TO THE NORMAL DOMAIN OF ATTRACTION OF A CAUCHY LAW

We consider the asymptotic null distribution of the empirical autocorrelation function when the innovations of a moving average process belong to the normal domain of attraction of a Cauchy law. A series expansion for the density of the limiting null distribution is developed, and some critical values of the tests are computed numerically.

Series representation for operator semistable laws and domains of normal attraction

The goal of this paper is to extend the series representation to operator semistable laws and to give necessary and sufficient conditions for a vector X to belong to the generalized domain of normal attraction of some operator semistable law having no Gaussian component.

Some remarks on the rate of convergence to stable laws

This note addresses the following question. Suppose we have a sequence of distributions of sums of independent identically distributed (i.i.d.) random summands converging to a given stable law. What properties of the summands have the most influence on the rate of convergence under consideration? At once it is necessary to note that a general answer is well known at present (a good reference for the accuracy of approximation with stable laws is the recent monograph [3], see also [6]-[11]), and it can be roughly stated that the existence of pseudomoments of higher order ensures better rates of convergence. In turn, the existence of pseudomoments depends on how close are the distribution of the summand and the stable law. But contrary to the case of the Gaussian limit law, where the existence of moments (or more precisely, tail behavior of the distribution) defines the rate of convergence and necessary and sufficient conditions for a given rate of convergence can be formulated, there is no such natural measure of closeness of the distributions in the case of the stable limit law. Taking the particular form of the density of the summand, namely, the Paretian distribution, which can be regarded in some sense as the "main" distribution in the domain of normal attraction (DNA) of a given stable distribution, we show how the rate of convergence depends on small perturbations of this density. Another goal of this note was to give an answer to the question raised in [1]. In that paper it was noted that the convergence to a stable distribution may be extremely slow (this was shown by means of simulation). It turns out that in the situation described in that paper summands belong to the domain of attraction (DA) but do not belong to the DNA. Our example shows that in this case the rate of convergence is only of logarithmic rate. This means that although theoretically both DNA and DA are equally important, in practice, limit theorems in the case of summands in DA but not in DNA are less useful due to the extremely slow rate of convergence and the fact that even asymptotic expansions cannot help very much (see Propositions 2 and 3, where new types of asymptotic expansions are given).

A law of the iterated logarithm for semistable laws on vector spaces and homogeneous groups

Let X1, X2, ... be a sequence of independent and identically distributed (i.i.d.)Rd-valued random vectors distributed according to a full (B,c) semistable law without Gaussian component. Then the following law of the iterated logarithm holds. $$\mathop {\lim \sup }\limits_{n \to \infty } \left\| {B^{ - (n + [\log n/\log c])} \sum\limits_{i = 1}^{[c^n ]} {X_i } } \right\|^{1/\log n} = 1 a.s.$$ This result is new even in the one-dimensional situation of semistable laws on the real line, where we extend our result to laws in the domain of normal attraction of a semistable law. Furthermore, we prove that this kind of law of the iterated logarithm also holds for certain semistable laws on homogeneous groups, especially on Heisenberg groups.

Moments of Limits of Lightly Trimmed Sums of Random Vectors in the Generalized Domain of Normal Attraction of Non-Gaussian Operator-Stable Laws

Existence and nonexistence for moments of limiting random vectors of normalized, lightly trimmed sums of random vectors in the generalized domain of normal attraction of non-Gaussian operator-stable laws are studied. The idea of representing the limiting random vectors by infinite series is essentially used in the proofs.

Central limit theorems for random processes with sample paths in exponential Orlicz spaces

This paper is devoted to the study of central limit theorems and the domain of normal attraction for some random processes with sample paths in exponential Orlicz spaces under metric entropy conditions. In particular, the local times of strongly symmetric standard Markov processes and random Fourier series are discussed.

Central limit theorem for stochastically continuous processes. Convergence to stable limit

LetX={X(t), t∈[0, 1]} be a stochastically continuous cadlag process. Assume that thek dimensional finite joint distributions ofX are in the domain of normal attraction of strictlyp-stable, 0 g(u−s) andΛp(|X(s−X(t|)∧|X(t)−X(u|)>f(u−s), 0 ≤s ≤t ≤u ≤ 1, conditions are found which imply that the distributions −(n−1/p(X1+···+Xn )),n≥1, converge weakly inD[0, 1] to the distribution of ap-stable process. HereX1,X2, ... are independent copies ofX andΛp(Z)=supt<0tpP{|Z|<t} denotes the weakpth moment of a random variable Z.

Domains of attraction of stable semigroups on simply connected nilpotent Lie groups

The classical Doeblin-Gnedenko conditions characterizing the domain of attraction of a non-Gaussian stable law, which have been proved to be sufficient for stable semigroups on simply connected nilpotent Lie groups by Carnal, are shown to be also necessary in that case. We also mention some consequences for the domain of normal attraction.

A CLT When Summands Come Randomly from r Populations

Let { Xn} be a sequence of mutually independent random variables (r.v.s.) defined on a probability space ( Ω, β, P) with respective distribution functions (d.f.s.) { Fn }, all of which belong to the domain of normal attraction of a symmetric stable law with characteristic exponent a, 0 &lt; a⩽2. Suppose further that at most r of the d.f.s. { Fn } are distinct, i.e. Fn ε { G1, G2,... Gr}. A Central Limit Theorem type result is proved when the number of variables among { X1 , X2, ... , Xn} that follow a Gj is a r.v. satisfying certain conditions.

Almost sure and weak invariance principles for random variables attracted by a stable law

Let (Xn) be i.i.d. random variables belonging to the domain of normal attraction of a symmetric stable law with parameter 0<p<2. We study the a.s. and weak approximation of the partial sum process\(S(t) - \sum\limits_{n \leqq t} {X_n (t \geqq } 0)\) by a symmetric stable processGp(t). Stout proved an upper bound for the optimal remainder term in this approximation; we prove here a lower bound, leaving only a small gap between the upper and lower estimates. We also give a new method to obtain upper bounds. Finally, we prove analogues of these results in the case when a.s. approximation is replaced by approximation in probability.

Operator stable laws: Series representations and domains of normal attraction

LetX,X 1,X 2,... be i.i.d. random vectors in ℝd. The limit laws μ that can arise by suitable affine normalizations of the partial sums,S n=X 1+...+X n, are calledoperator-stable laws. These laws are a natural extension to ℝ d of the stable laws onℝ. Thegeneralized domain of attraction of μ[GDOA(μ)] is comprised of all random vectorsX whose partial sums can be affinely normalized to converge to μ. If the linear part of the affine transformation is restricted to take the formn −B for some exponent operatorB naturally associated to μ thenX is in thegeneralized domain of normal attraction of μ [GDONA(μ)]. This paper extends the theory of operator-stable laws μ and their domains of attraction and normal attraction.

Marches aléatoires récurrentes totalement dissymétriques et domaine d⊃attraction des lois stables

The aim of this article is to answer some question raised by A. Brunei and D. Revuz concerning the Martin boundary for recurrent random walks on groups.Let G be a group and a the recurrent potential measure of a random walk on We characterise those random walks for which lim infx , x a{x) is finite, showing that this property is related to the renewal theory. As an example,we prove that, on Z, the walks in the domain of normal attraction of any centered and maximally asymmetric stable walk with parameter ae]l,2[ share this property

A local limit theorem for attractions under a stable law

AbstractLet {Xn} be a sequence of independent random variables each having a common d.f. V1. Suppose that V1 belongs to the domain of normal attraction of a stable d.f. V0 of index α 0 ≤ α ≤ 2. Here we prove that, if the c.f. of X1 is absolutely integrable in rth power for some integer r &gt; 1, then for all large n the d.f. of the normalized sum Zn of X1, X2, …, Xn is absolutely continuous with a p.d.f. vn such thatas n → ∞, where v0 is the p.d.f. of Vo.