Sequence Of Dependent Random Variables Research Articles

Nonparametric estimation of the relative error regression for twice censored and dependent data

Let ( Y i ) 1 ≤ i ≤ n be a sequence of dependent random variables (r.v.) of interest distributed as Y and ( X i ) 1 ≤ i ≤ n be a corresponding d−dimensional vector of covariates taking values on ℝ d . The r.v. Y is twice censored and satisfies the α−mixing property. In this article, we are interested in the relative error regression estimation in the case of twice-censored data under strong mixing conditions. Under suitable assumptions, the estimator’s strong uniform almost sure consistency with rate and asymptotic normality are established. We have highlighted the covariance term, which is relatively uncommon in this context. Furthermore, we give a result about the α-mixing rate in the case of twice-censoring. As far as we know, there is no analogous result with proof in the context of twice-censoring. A simulation study is conducted in both one and two dimensions for the covariate to highlight the performance of our estimator.

NEW TAIL PROBABILITY TYPE INEQUALITIES AND COMPLETE CONVERGENCE FOR WOD RANDOM VARIABLES WITH APPLICATION TO LINEAR MODEL GENERATED BY WOD ERRORS

In this paper, we prove an exponential type inequality for unbounded Widely Orthant Dependent (WOD) random variables. Then we will use this inequality for establishing the almost complete convergence for a sequence of widely dependent random variables (WOD). As an application, we combine Markov inequalities and the new exponential inequality to discuss almost complete convergence of errors generated by a sequence of dependent random variables (WOD) in linear models.

Moderate deviation principle for $ m $-dependent random variables under the sub-linear expectation

<abstract><p>Let $ \{X_n, n\geq1\} $ be a sequence of $ m $-dependent strictly stationary random variables in a sub-linear expectation $ (\Omega, \mathcal{H}, \mathbb{E}) $. In this article, we give the definition of $ m $-dependent sequence of random variables under sub-linear expectation spaces taking values in $ \mathbb{R} $. Then we establish moderate deviation principle for this kind of sequence which is strictly stationary. The results in this paper generalize the result that in the case of independent identically distributed samples. It provides a basis to discuss the moderate deviation principle for other types of dependent sequences.</p></abstract>

Strong Limit Theorems for Dependent Random Variables

In this article We establish moment inequality of dependent random variables, furthermore some theorems of strong law of large numbers and complete convergence for sequences of dependent random variables. In particular, independent and identically distributed Marcinkiewicz Law of large numbers are generalized to the case of m₀ -dependent sequences.

Asymptotic Approximations of Ratio Moments Based on Dependent Sequences

The widely orthant dependent (WOD) sequences are very weak dependent sequences of random variables. For the weighted sums of non-negative m-WOD random variables, we provide asymptotic expressions for their appropriate inverse moments which are easy to calculate. As applications, we also obtain asymptotic expressions for the moments of random ratios. It is pointed out that our random ratios can include some models such as change-point detection. Last, some simulations are illustrated to test our results.

Functional CLT for nonstationary strongly mixing processes

This paper deals with the functional central limit theorem for non-stationary dependent sequences of random variables satisfying the Lindeberg condition. The dependence condition which we impose is known under the name of weak strong mixing condition. It is satisfied by a large class of dependent random variables, including functions of strongly mixing or α-dependent Markov chains.

Almost sure central limit theorems for weighted dependent sequences

Abstract Let {Xn: n ≧ 1} be a sequence of dependent random variables and let {wnk: 1 ≦ k ≦ n, n ≧ 1} be a triangular array of real numbers. We prove the almost sure version of the CLT proved by Peligrad and Utev [7] for weighted partial sums of mixing and associated sequences of random variables, i.e.

On the law of the iterated logarithm for sequences of dependent random variables

Sufficient conditions for the applicability of the law of the iterated logarithm to sequences of dependent random variables are obtained. As a corollary, a theorem on the law of the iterated logarithm for a sequence of m-orthogonal random variables is proved.

On the law of the iterated logarithm for sequences of dependent random variables

On the law of the iterated logarithm for sequences of dependent random variables

The strong law of large numbers for a stationary sequence

General results on the applicability of the strong law of large numbers to a sequence of dependent random variables, as formulated in terms of estimates for the moments of sums of such variables, are applied to give new conditions of the applicability of this law to (in a wide sense) a stationary sequence of random variables.

On the strong law of large numbers for a stationary sequence

General results on the applicability of the strong law of large numbers to a sequence of dependent random variables, as formulated in terms of estimates for the moments of sums of such variables, are applied to give new conditions of the applicability of this law to (in a wide sense) a stationary sequence of random variables.

On stochastic dominance and the strong law of large numbers for dependent random variables

We study sequences of dependent random variables which in a sense are dominated by a sequence of independent random variables. The introduced concept unifies the notion of widely upper (lower) orthant dependent sequences, weak negative dependence and some stochastic orders. For some class of dependent sequences we prove the strong law of large numbers.

Estimates for the ruin probability of a time-dependent renewal risk model with dependent by-claims

Consider a continuous-time renewal risk model, in which every main claim induces a delayed by-claim. Assume that the main claim sizes and the inter-arrival times form a sequence of identically distributed random pairs, with each pair obeying a dependence structure, and so do the by-claim sizes and the delay times. Supposing that the main claim sizes with by-claim sizes form a sequence of dependent random variables with dominatedly varying tails, asymptotic estimates for the ruin probability of the surplus process are investigated, by establishing a weakly asymptotic formula, as the initial surplus tends to infinity.

Large deviation for negatively dependent random variables under sublinear expectation

In this article, first we give the definition of negatively dependent sequence of random variables under sublinear expectation then we establish large deviation principle for this kind of sequence. Moreover, we obtain the upper bound of moderate deviation principle.

On the Strong Law of Large Numbers for Sequences of Dependent Random Variables with Finite Second Moments

New sufficient conditions of a.s. convergence of the series $$ {\displaystyle \sum_{n=1}^{\infty }{X}_n} $$ and new sufficient conditions for the applicability of the strong law of large numbers are established for a sequence of dependent random variables {X n } = 1 ∞ with finite second moments. These results are generalizations of the well-known theorems on a.s. convergence of the series of orthogonal random variables and on the strong law of large numbers for orthogonal random variables (Men’shov–Rademacher and Petrov’s theorems). It is shown that some of the results obtained are optimal.

Asymptotic normality of DHD estimators in a partially linear model

The paper studies a partially linear regression model given by $$\begin{aligned} y_i=x_i^T\beta +f(t_i)+\varepsilon _i,i=1,2,\ldots ,n, \end{aligned}$$ where \(\{\varepsilon _i,i=1,2,\ldots , n\}\) are independent and identically distributed random errors with zero mean and finite variance \(\sigma ^2>0\). Using a difference based and the Huber–Dutter (DHD) approaches, the estimators of unknown parametric component \(\beta \) and root variance \(\sigma \) are given, and then the estimation of nonparametric component \(f(\cdot )\) is given by the wavelet method. The asymptotic normality of the DHD estimators of \(\beta \) and \(\sigma \) are investigated, and the weak convergence rate of the estimator of \(f(\cdot )\) is also investigated. In addition, for stationary \(m\)-dependent sequence of random variables, the central limit theorem is also obtained. At last, two examples are presented to illustrate the proposed method.

A Note on Exponential Inequalities of ψ-Weakly Dependent Sequences

Two exponential inequalities are established for a wide class of general weakly dependent sequences of random variables, called <TEX>${\psi}$</TEX>-weakly dependent process which unify weak dependence conditions such as mixing, association, Gaussian sequences and Bernoulli shifts. The <TEX>${\psi}$</TEX>-weakly dependent process includes, for examples, stationary ARMA processes, bilinear processes, and threshold autoregressive processes, and includes essentially all classes of weakly dependent stationary processes of interest in statistics under natural conditions on the process parameters. The two exponential inequalities are established on more general conditions than some existing ones, and are proven in simpler ways.

On the Strong Law of Large Numbers for a Sequence of Dependent Random Variables

New sufficient conditions are found for the applicability of the strong law of large numbers to a sequence of random variables without assumptions of independence or nonnegativity. Bibliography: 3 titles.

System availability behavior of some stationary dependent sequences

We consider the system availability behavior of a one-unit repairable system when the failure and the repair times are generated by a stationary dependent sequence of random variables. We obtain the general expression for the point availability, and discuss the nature of the availability measure for two time series models: a first-order exponential moving average process and a first-order exponential autoregressive process.

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.