Rosenthal-type Inequality Research Articles

On complete convergence for weighted sums of m-widely acceptable random variables under sub-linear expectations and its statistical applications

. In this article, by means of Rosenthal-type inequality, we study the complete convergence and strong law of large numbers for weighted sums of m-widely accpetable random variables under sub-linear expectations, which extend some existing results in classical probability space. As applications, we investigate the complete consistency and strong consistency of weighted estimators in nonparametric regression models under sub-linear expectations. In addition, we provide some simulation studies to verify the finite sample performance of the theoretical results.

Complete convergence for weighted sums of random variables satisfying generalized Rosenthal-type inequalities*

Complete convergence for weighted sums of random variables satisfying generalized Rosenthal-type inequalities*

Complete convergence and complete moment convergence for weighted sums of random variables satisfying generalized Rosenthal type inequalities and an application

This article mainly investigates the complete convergence and complete moment convergence for weighted sums of random variables satisfying generalized Rosenthal type inequalities, which improves the corresponding ones of the literature. As an application, we further research the errors-in-variables (EV, for short) regression models and exhibit the strong consistency for the least square (LS, for short) estimators. In addition, simulation studies are conducted to verify the finite sample performance of theoretical results.

A Rosenthal-type Inequality for Some Classes of Dependent Random Variables and Its Applications

A Rosenthal-type Inequality for Some Classes of Dependent Random Variables and Its Applications

Weak convergence for weighted sums of a class of random variables with related statistical applications

In this paper, we study the weak convergence and convergence rate in the weak law of large numbers for weighted sums of a class of random variables satisfying the Rosenthal type inequality. The necessary and sufficient conditions for the convergence rates in the weak law of large numbers under some mild conditions are provided. Moreover, the main results that we established are applied to simple linear errors-in-variables regression models and nonparametric regression models based on a class of random errors. Finally, we present some numerical simulations to assess the finite sample performance of the theoretical results.

Convergence rates in the functional CLT for [formula omitted]-mixing triangular arrays

We obtain convergence rates (in the Levi–Prokhorove metric) in the functional central limit theorem (CLT) for partial sums Sn=∑j=1nξj,n of triangular arrays {ξ1,n,ξ2,n,…,ξn,n} satisfying some mixing and moment conditions (which are not necessarily uniform in n). For certain classes of additive functionals of triangular arrays of contracting Markov chains (in the sense of Dobrushin) we obtain rates which are close to the best rates obtained for independent random variables. In addition, we obtain close to optimal rates in the usual CLT and a moderate deviations principle and some Rosenthal type inequalities. We will also discuss applications to some classes of local statistics (e.g. covariance estimators), as well as expanding non-stationary dynamical systems, which can be reduced to non-uniformly mixing triangular arrays by an approximation argument. The main novelty here is that our results are obtained without any assumptions about the growth rate of the variance of Sn. The result are obtained using a certain type of block decomposition, which, in a sense, reduces the problem to the case when the variance of Sn is “not negligible” in comparison with the (new) number of summands.

Capacity inequalities and strong laws for m-widely acceptable random variables under sub-linear expectations

In this work, a generalised concept of dependent random variables, i.e., m-widely acceptable random variables, is put forward and some inequalities such as exponential inequality and Rosenthal-type inequality are established under sub-linear expectations. By virtue of these inequalities, we obtain a general result on complete convergence and the Marcinkiewicz-Zygmund type strong law of large numbers for weighted sums of m-widely acceptable random variables, which extend and improve some existing ones in classical probability space as well as in sub-linear expectation space. As applications of the main results, the complete consistency and strong consistency of weighted estimators in nonparametric regression model under sub-linear expectations are also investigated.

On the linearly extended negative quadrant dependent random variables and its inequalities

In this article, the structure of linearly extended negative quadrant dependent (LENQD) random variables is given, and its basic properties are investigated. In addition, the characteristic function inequality, Bernstein-type inequality, some probability inequalities, and the Rosenthal-type inequality are established for LENQD random variables.

The law of iterated logarithm for a class of random variables satisfying Rosenthal type inequality

<abstract><p>Let $ \{Y_n, n\geq 1\} $ be sequence of random variables with $ EY_n = 0 $ and $ \sup_nE|Y_n|^p < \infty $ for each $ p > 2 $ satisfying Rosenthal type inequality. In this paper, the law of the iterated logarithm for a class of random variable sequence with non-identical distributions is established by the Rosenthal type inequality and Berry-Esseen bounds. The results extend the known ones from i.i.d and NA cases to a class of random variable satisfying Rosenthal type inequality.</p></abstract>

On the error bound in the normal approximation for Jack measures

In this paper, we obtain uniform and non-uniform bounds on the Kolmogorov distance in the normal approximation for Jack deformations of the character ratio, by using Stein’s method and zero-bias couplings. Our uniform bound comes very close to that conjectured by Fulman (J. Combin. Theory Ser. A 108 (2004) 275–296). As a by-product of the proof of the non-uniform bound, we obtain a Rosenthal-type inequality for zero-bias couplings.

Laws of large numbers and complete convergence for WOD random variables and their applications

By using the Marcinkiewicz-Zygmund type inequality and Rosenthal type inequality, we study the Lr convergence, weak law of large numbers and the complete convergence for usual normed sums and weighted sums of arrays of rowwise widely orthant dependent (WOD, for short) random variables. In addition, some applications of the Lr convergence and the complete convergence to nonparametric regression models based on WOD errors are provided. At last, we provide a numerical simulation to verify the validity of our theoretical result.

Strong laws for weighted sums of random variables satisfying generalized Rosenthal type inequalities

Let 1le p<2 and 0<alpha , beta <infty with 1/p=1/alpha +1/beta . Let {X_{n}, nge 1} be a sequence of random variables satisfying a generalized Rosenthal type inequality and stochastically dominated by a random variable X with E|X|^{beta }< infty . Let {a_{nk}, 1le kle n, nge 1} be an array of constants satisfying sum_{k=1}^{n} |a_{nk}|^{alpha }=O(n). Marcinkiewicz–Zygmund type strong laws for weighted sums of the random variables are established. Our results generalize or improve the corresponding ones of Wu (J. Inequal. Appl. 2010:383805, 2010), Huang et al. (J. Math. Inequal. 8:465–473, 2014), and Wu et al. (Test 27:379–406, 2018).

Nonconventional moderate deviations theorems and exponential concentration inequalities

Nous obtenons un theoreme de deviations moderees et des inegalites de concentration exponentielles (du type de Bernstein) pour des sommes «non-conventionnelles» de la forme $S_{N}=\sum_{n=1}^{N}(F(\xi_{q_{1}(n)},\xi_{q_{2}(n)},\ldots,\xi_{q_{\ell}(n)})-\bar{F})$, ou la plupart du temps nous considerons $q_{i}(n)=in$, mais nos resultats restent aussi vrais pour des $q_{i}(n)$ plus generaux tels que des polynomes. Ici, $\xi_{n}$, $n\geq 0$ est un processus vectoriel suffisamment melangeant avec des conditions de stationnarite, $F$ est une fonction satisfaisant certaines proprietes de regularite et $\bar{F}$ est une constante de centrage. Quand $\xi_{n}$, $n\geq 0$ sont independants et identiquement distribues, un principe de grande deviation a ete obtenu dans (Probab. Theory Related Fields 158 (2014) 197–224) et un des objectifs de cet article est d’obtenir des resultats analogues dans le cas faiblement dependant. Plusieurs resultats de type approximation normale sont aussi obtenus. En particulier, deux nouvelles preuves du theoreme central limite non-conventionnel sont donnees et une inegalite de type Rosenthal est obtenue. Nos resultats sont vrais par exemple quand $\xi_{n}=(T^{n}f_{i})_{i=1}^{\wp}$ ou $T$ est un sous-shift de type fini topologiquement melangeant, une application Gibbs–Markov, un diffeomorphisme hyperbolique, une tour de Young ou une transformation expansive pour une mesure invariante de Gibbs, tout comme dans le cas ou $\xi_{n}$, $n\geq 0$ forme une suite stationnaire exponentiellement (ou streched exponentiellement) $\phi$-melangeante, ce qui, par exemple, et vrai lorsque $\xi_{n}=(f_{i}(\Upsilon_{n}))_{i=1}^{\wp}$ ou $\Upsilon_{n}$ est une chaine de Markov satisfaisant une condition de Doeblin, consideree comme un processus stationnaire par rapport a une mesure invariante.

Lq‐valued Burkholder–Rosenthal inequalities and sharp estimates for stochastic integrals

AbstractWe prove sharp maximal inequalities for ‐valued stochastic integrals with respect to any Hilbert space‐valued local martingale. Our proof relies on new Burkholder–Rosenthal type inequalities for martingales taking values in an ‐space.

On consistency of wavelet estimator in nonparametric regression models

In this paper, we mainly investigate the nonparametric regression model with repeated measurements based on extended negatively dependent (END, in short) errors. Based on the Rosenthal type inequality and the Marcinkiewicz–Zygmund type strong law of large numbers, the mean consistency, weak consistency, strong consistency, complete consistency and strong convergence rate of the wavelet estimator are established under some mild conditions, which generalize the corresponding ones for negatively associated errors. Some numerical simulations are presented to verify the validity of the theoretical results.

On the martingale central limit theorem for strictly stationary sequences

In this study, for strictly stationary martingale difference sequences, we completely characterize the domain of attraction for the standard normal law (a' la Lévy, Khinchine, and Feller). In particular, we obtain the Rosenthal type inequality (Theorem 4) and a new relative stability result (Theorem 2).

Partial quasi-likelihood analysis

The quasi-likelihood analysis is generalized to the partial quasi-likelihood analysis. Limit theorems for the quasi-likelihood estimators, especially the quasi-Bayesian estimator, are derived in the situation where existence of a slow mixing component prohibits the Rosenthal type inequality from applying to the derivation of the polynomial type large deviation inequality for the statistical random field. We give two illustrative examples.

L_{p}-convergence, complete convergence, and weak laws of large numbers for asymptotically negatively associated random vectors with values in mathbb{R}^{d}

In this paper, based on the Rosenthal-type inequality for asymptotically negatively associated random vectors with values in mathbb{R}^{d}, we establish results on L_{p}-convergence and complete convergence of the maximums of partial sums are established. We also obtain weak laws of large numbers for coordinatewise asymptotically negatively associated random vectors with values in mathbb{R}^{d}.

The asymptotic normality of the linear weighted estimator in nonparametric regression models

ABSTRACTConsider the following nonparametric regression model:where xni are known fixed design points from for some positive integer d ⩾ 1, g( · ) is an unknown regression function defined on A and ϵni are random errors. Under some suitable conditions, the asymptotic normality of the linear weighted estimator of g in the nonparametric regression model based on ρ-mixing errors is established. The key techniques used in the paper are the Rosenthal type inequality and the Bernstein’s bigblock and small-block procedure. The result obtained in the paper generalizes the corresponding ones for some dependent sequences.

An alternative to the coupling of Berkes–Liu–Wu for strong approximations

In this paper we propose an alternative to the coupling of Berkes, Liu and Wu [1] to obtain strong approximations for partial sums of dependent sequences. The main tool is a new Rosenthal type inequality expressed in terms of the coupling coefficients. These coefficients are well suited to some classes of Markov chains or dynamical systems, but they also give new results for smooth functions of linear processes.