Limit Theorem For Triangular Arrays Research Articles

MỘT SỐ ĐỊNH LÝ VỀ SỰ HỘI TỤ THEO TRUNG BÌNH CỦA MẢNG TAM GIÁC CÁC BIẾN NGẪU NHIÊN mn-PHỤ THUỘC ĐÔI MỘT THEO HÀNG

This paper establishes a mean convergence theorem for triangular arrays of rowwise and pairwise mn-dependent random variables. Some authors studied limit theorems for sequences of pairwise m-dependent random variables where m is fixed (see, e.g., Quang and Nguyen [Applications of Mathematics, 2016] and Thanh [Bulletin of the Institute of Mathematics Academia Sinica, 2005]). In this paper, we establish a limit theorem for triangular arrays of rowwise and pairwise mn-dependent random variables, where mn may approach infinity as n → ∞. The main theorem extends some results in the literature, including Theorem 3.1 of Chen, Bai and Sung in [Journal of Mathematical Analysis and Applications, 2014].

Superconvergence in free probability limit theorems for arbitrary triangular arrays

It is known that limit theorems for triangular arrays with identically distributed rows yields convergence of densities rather than just convergence in distribution. We show that this superconvergence result holds—at least at points at which the limit density is nonzero—even if the rows of the array are not identically distributed.

On central limit theorems for IV-events

Interval-valued fuzzy sets were introduced in 1970s as an extension of Zadeh’s fuzzy sets. For interval-valued fuzzy events, (IV-events for short) IV-probability theory has been developed. In this paper, we prove central limit theorems for triangular arrays of IV-observables within this theory. We prove the Lindeberg CLT and the Lyapunov CLT, assuming that IV-observables are not necessary identically distributed. We also prove the Feller theorem for null arrays of IV-observables. Furthermore, we present examples of applications of the aforementioned theorems. In particular, we study the convergence in distribution of scaled sums of identically distributed IV-observables.

LAN property for a simple Lévy process

In this paper, we consider a simple Lévy process given by a Brownian motion and a compensated Poisson process, whose drift and diffusion parameters as well as its intensity are unknown. Supposing that the process is observed discretely at high frequency, we derive the local asymptotic normality (LAN) property. In order to obtain this result, Malliavin calculus and Girsanov's theorem are applied in order to write the log-likelihood ratio in terms of sums of conditional expectations, for which a central limit theorem for triangular arrays can be applied.

A central limit theorem for triangular arrays of weakly dependent random variables, with applications in statistics

We derive a central limit theorem for triangular arrays of possibly nonstationary random variables satisfying a condition of weak dependence in the sense of Doukhan and Louhichi [Stoch. Proc. Appl. 84 (1999) 313–342]. The proof uses a new variant of the Lindeberg method: the behavior of the partial sums is compared to that of partial sums of dependent Gaussian random variables. We also discuss a few applications in statistics which show that our central limit theorem is tailor-made for statistics of different type.

Moment bounds and central limit theorems for Gaussian subordinated arrays

A general moment bound for sums of products of Gaussian vector’s functions extending the moment bound in Taqqu (1977, Lemma 4.5) [28] is established. A general central limit theorem for triangular arrays of nonlinear functionals of multidimensional non-stationary Gaussian sequences is proved. This theorem extends the previous results of Breuer and Major (1983) [5], Arcones (1994) [1] and others. A Berry–Esseen-type bound in the above-mentioned central limit theorem is derived following Nourdin et al. (2011) [20]. Two applications of the above results are discussed. The first one refers to the asymptotic behavior of a roughness statistic for continuous-time Gaussian processes and the second one is a central limit theorem satisfied by long memory locally stationary processes.

Negative Dependence in Sampling

Abstract. The strong Rayleigh property is a new and robust negative dependence property that implies negative association; in fact it implies conditional negative association closed under external fields (CNA+). Suppose that and are two families of 0‐1 random variables that satisfy the strong Rayleigh property and let . We show that {Zi} conditioned on is also strongly Rayleigh; this turns out to be an easy consequence of the results on preservation of stability of polynomials of Borcea & Brändén (Invent. Math., 177, 2009, 521–569). This entails that a number of important πps sampling algorithms, including Sampford sampling and Pareto sampling, are CNA+. As a consequence, statistics based on such samples automatically satisfy a version of the Central Limit Theorem for triangular arrays.

A note on two measures of dependence

In a paper in 1956, Dobrushin proved a central limit theorem for triangular arrays of Markov chains under certain dependence assumptions somewhat related to “Doeblin’s condition”. In a very recent paper, Peligrad proved a central limit theorem of the same type, but with dependence assumptions based on maximal correlations. This note here will give a concrete example to illustrate the extent to which Peligrad’s result goes beyond that of Dobrushin.

Central limit theorem for triangular arrays of non-homogeneous Markov chains

In this paper we obtain the central limit theorem for triangular arrays of non-homogeneous Markov chains under a condition imposed to the maximal coefficient of correlation. The proofs are based on martingale techniques and a sharp lower bound estimate for the variance of partial sums. The results complement an important central limit theorem of Dobrushin based on the contraction coefficient.

Goodness-of-fit tests for Markovian time series models: Central limit theory and bootstrap approximations

New goodness-of-fit tests for Markovian models in time series analysis are developed which are based on the difference between a fully nonparametric estimate of the one-step transition distribution function of the observed process and that of the model class postulated under the null hypothesis. The model specification under the null allows for Markovian models, the transition mechanisms of which depend on an unknown vector of parameters and an unspecified distribution of i.i.d. innovations. Asymptotic properties of the test statistic are derived and the critical values of the test are found using appropriate bootstrap schemes. General properties of the bootstrap for Markovian processes are derived. A new central limit theorem for triangular arrays of weakly dependent random variables is obtained. For the proof of stochastic equicontinuity of multidimensional empirical processes, we use a simple approach based on an anisotropic tiling of the space. The finite-sample behavior of the proposed test is illustrated by some numerical examples and a real-data application is given.

Moment bounds and central limit theorem for functions of Gaussian vectors

In this note, bounds for moments of functions of Gaussian vectors are proved, generalizing earlier results by Taqqu (Z. Wahrscheinlichkeitstheorie verw. Gebite 40 (1977) 203) and Arcones (Ann. probab. 15 (4) (1994) 2243). These bounds are used to derive a Lindeberg–Levy central limit theorem for triangular arrays of functions of Gaussian vectors. Statistical applications for long range dependent processes are given.

A triangular central limit theorem under a new weak dependence condition

We use a new weak dependence condition from Doukhan and Louhichi (Stoch. Process. Appl. 1999, 84, 313–342) to provide a central limit theorem for triangular arrays; this result applies for linear arrays (as in Peligrad and Utev, Ann. Probab. 1997, 25(1), 443–456) and standard kernel density estimates under weak dependence. This extends on strong mixing and includes non-mixing Markov processes and associated or Gaussian sequences. We use Lindeberg method in Rio (Probab. Theory Related Fields 1996, 104, 255–282).

Asymptotic Normality for Density Kernel Estimators in Discrete and Continuous Time

In this paper, we build a central limit theorem for triangular arrays of sequences which satisfy a mild mixing condition. This result allows us to study asymptotic normality of density kernel estimators for some classes of continuous and discrete time processes.

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.

Central Limit Theorems for Dependent Heterogeneous Random Variables

This paper presents central limit theorems for triangular arrays of mixingale and near-epoch-dependent random variables. The central limit theorem for near-epoch-dependent random variables improves results from the literature in various respects. The approach is to define a suitable Bernstein blocking scheme and apply a martingale difference central limit theorem, which in combination with weak dependence conditions renders the result. The most important application of this central limit theorem is the improvement of the conditions that have to be imposed for asymptotic normality of minimization estimators.

Central Limit Theorems for Markov‐Connected Random Variables

AbstractThe 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 (Do). One result represents a central limit theorem without the assumption of finite variances.

Limit theorems for triangular arrays under a relaxed asymptotic negligibility condition

AbstractLet {Xnk} be a triangular array of independent random variables satisfying the so-called tail-negligibility condition, i.e. such that Prob{|Xnk| > a} → 0 as both k, n → ∞. It is also assumed that for each fixed k, Xnk converges in distribution as n → ∞. Theorems on the asymptotic behavior of the row sums of the array, analogous to those of the classical theory under the uniform negligibility condition, are presented.

A central limit theorem for D( A)-valued processes

Let D ( A ) be the space of set-indexed functions that are outer continuous with inner limits, a generalization of D[0, 1]. This paper proves a central limit theorem for triangular arrays of independent D ( A ) valued random variables. The limit processes are not restricted to be Gaussian, but can be quite general infinitely divisible processes. Applications of the theorem include construction of set-indexed Lévy processes and a unified central limit theorem for partial sum processes and generalized empirical processes. Results obtained are new even for the D[0, 1] case.