Marcinkiewicz-Zygmund Type Strong Law Research Articles

Equivalent conditions of the complete convergence for weighted sums of NSD random variables

In this paper, we consider the complete convergence for weighted sums of negatively superadditive-dependent (NSD) random variables without assumptions of identical distribution. Some sufficient and necessary conditions to prove the complete convergence for weighted sums of NSD random variables are presented, which extend and improve the corresponding ones of Naderi et al. As an application of the main results, the Marcinkiewicz–Zygmund type strong law of large numbers for weighted sums of NSD random variables is also achieved.

Gaussian polytopes: A cumulant-based approach

The random convex hull of a Poisson point process in Rd whose intensity measure is a multiple of the standard Gaussian measure on Rd is investigated. The purpose of this paper is to invent a new viewpoint on these Gaussian polytopes that is based on cumulants and the general large deviation theory of Saulis and Statulevičius. This leads to new and powerful concentration inequalities, moment bounds, Marcinkiewicz–Zygmund-type strong laws of large numbers, central limit theorems and moderate deviation principles for the volume and the face numbers. Corresponding results are also derived for the empirical measures induced by these key geometric functionals, taking thereby care of their spatial profiles.

On the strong convergence properties for weighted sums of negatively orthant dependent random variables

In the paper, the strong convergence properties for two different weighted sums of negatively orthant dependent (NOD) random variables are investigated. Let {X n , n ≥ 1} be a sequence of NOD random variables. The results obtained in the paper generalize the corresponding ones for i.i.d. random variables and identically distributed NA random variables to the case of NOD random variables, which are stochastically dominated by a random variable X. As a byproduct, the Marcinkiewicz-Zygmund type strong law of large numbers for NOD random variables is also obtained.

Complete Convergence and Complete Moment Convergence for Maximal Weighted Sums of Extended Negatively Dependent Random Variables

In this paper, the complete convergence and complete moment convergence for maximal weighted sums of extended negatively dependent random variables are investigated. Some sufficient conditions for the convergence are provided. In addition, the Marcinkiewicz–Zygmund type strong law of large numbers for weighted sums of extended negatively dependent random variables is obtained. The results obtained in the article extend the corresponding ones for independent random variables and some dependent random variables.

Lp convergence and complete convergence for weighted sums of AANA random variables

ABSTRACTLet {Xn, n ⩾ 1} be a sequence of asymptotically almost negatively associated (AANA, for short) random variables which is stochastically dominated by a random variable X, and {dni, 1 ⩽ i ⩽ n, n ⩾ 1} be a sequence of real function, which is defined on a compact set E. Under some suitable conditions, we investigate some convergence properties for weighted sums of AANA random variables, especially the Lp convergence and the complete convergence. As an application, the Marcinkiewicz–Zygmund-type strong law of large numbers for AANA random variables is obtained.

Baum–Katz type theorems with exact threshold

Let be either a sequence of arbitrary random variables, or a martingale difference sequence, or a centred sequence with a suitable level of negative dependence. We prove Baum–Katz type theorems by only assuming that the variables satisfy a uniform moment bound condition. We also prove that this condition is best possible even for sequences of centred, independent random variables. This leads to Marcinkiewicz–Zygmund type strong laws of large numbers with estimate for the rate of convergence.

Weighted version of strong law of large numbers for a class of random variables and its applications

In this paper, the single index weighted version of Marcinkiewicz–Zygmund type strong law of large numbers and the double index weighted version of Marcinkiewicz–Zygmund type strong law of large numbers are investigated successively for a class of random variables, which extends the classical results for independent and identically distributed random variables. As applications of the results, we further study the strong consistency for the weighted estimator in the nonparametric regression model and the least square estimators in the simple linear errors-in-variables model. Moreover, we also present some numerical study to verify the validity of our results.

ON COMPLETE CONVERGENCE AND COMPLETE MOMENT CONVERGENCE FOR A CLASS OF RANDOM VARIABLES

In this paper, the complete convergence and complete moment convergence for a class of random variables satisfying the Rosenthal type inequality are investigated. The sufficient and necessary conditions for the complete convergence and complete moment convergence are provided. As applications, the Baum-Katz type result and the Marcinkiewicz-Zygmund type strong law of large numbers for a class of random variables satisfying the Rosenthal type inequality are established. The results obtained in the paper extend the corresponding ones for some dependent random variables.

Complete moment convergence for weighted sums of negatively orthant dependent random variables

In this paper, the complete moment convergence and the integrability of the supremum for weighted sums of negatively orthant dependent (NOD, in short) random variables are presented. As applications, the complete convergence and the Marcinkiewicz-Zygmund type strong law of large numbers for NODrandom variables are obtained. The results established in the paper generalize some corresponding ones for independent random variables and negatively associated random variables.

Strong and weak consistency of LS estimators in the EV regression model with negatively superadditive-dependent errors

In this paper, the strong laws of large numbers for partial sums and weighted sums of negatively superadditive-dependent (NSD, in short) random variables are presented, especially the Marcinkiewicz–Zygmund type strong law of large numbers. Using these strong laws of large numbers, we further investigate the strong consistency and weak consistency of the LS estimators in the EV regression model with NSD errors, which generalize and improve the corresponding ones for negatively associated random variables. Finally, a simulation is carried out to study the numerical performance of the strong consistency result that we established.

Complete convergence for weighted sums of END random variables and its application to nonparametric regression models

ABSTRACTIn this article, the complete convergence for weighted sums of extended negatively dependent (END, for short) random variables is investigated. Some sufficient conditions for the complete convergence are provided. In addition, the Marcinkiewicz–Zygmund type strong law of large numbers for weighted sums of END random variables is obtained. The results obtained in the article generalise and improve the corresponding one of Wang et al. [(2014b), ‘On Complete Convergence for an Extended Negatively Dependent Sequence’, Communications in Statistics-Theory and Methods, 43, 2923–2937]. As an application, the complete consistency for the estimator of nonparametric regression model is established.

On the complete convergence of weighted sums for an array of rowwise negatively superadditive dependent random variables

In this paper, the complete convergence and the complete moment convergence of weighted sums for an array of negatively superadditive dependent random variables are established. The results generalize the Baum-Katz theorem on negatively superadditive dependent random variables. In particular, the Marcinkiewicz-Zygmund type strong law of large numbers of weights sums for sequences of negatively superadditive dependent random variables is obtained.

General results of complete convergence and complete moment convergence for weighted sums of some class of random variables

ABSTRACTIn the article, the complete convergence and complete moment convergence for weighted sums of sequences of random variables satisfying a maximal Rosenthal type inequality are studied. As an application, the Marcinkiewicz–Zygmund type strong law of large numbers is obtained. Our partial results generalize and improve the corresponding ones of Shen (2013).

Exponential probability inequalities for WNOD random variables and their applications

Some exponential probability inequalities for widely negative orthant dependent (WNOD, in short) random variables are established, which can be treated as very important roles to prove the strong law of large numbers among others in probability theory and mathematical statistics. By using the exponential probability inequalities, we study the complete convergence for arrays of rowwise WNOD random variables. As an application, the Marcinkiewicz–Zygmund type strong law of large numbers is obtained. In addition, the complete moment convergence for arrays of rowwise WNOD random variables is studied by using the exponential probability inequality and complete convergence that we established.

Complete moment convergence of pairwise NQD random variables

It is known that the dependence structure of pairwise negative quadrant dependent (NQD) random variables is weaker than those of negatively associated random variables and negatively orthant dependent random variables. In this article, we investigate the moving average process which is based on the pairwise NQD random variables. The complete moment convergence and the integrability of the supremum are presented for this moving average process. The results imply complete convergence and the Marcinkiewicz–Zygmund-type strong law of large numbers for pairwise NQD sequences.

Complete convergence of weighted sums for arrays of rowwise φ-mixing random variables

In this paper, we establish the complete convergence and complete moment convergence of weighted sums for arrays of rowwise φ-mixing random variables, and the Baum-Katz-type result for arrays of rowwise φ-mixing random variables. As an application, the Marcinkiewicz-Zygmund type strong law of large numbers for sequences of φ-mixing random variables is obtained. We extend and complement the corresponding results of X. J. Wang, S. H. Hu (2012).

On the laws of large numbers for double arrays of independent random elements in Banach spaces

For a double array of independent random elements {V mn ,m ≥ 1, n ≥ 1} in a real separable Banach space, conditions are provided under which the weak and strong laws of large numbers for the double sums Σ i=1 m Σ j=1 n V ij , m ≥ 1, n ≥ 1 are equivalent. Both the identically distributed and the nonidentically distributed cases are treated. In the main theorems, no assumptions are made concerning the geometry of the underlying Banach space. These theorems are applied to obtain Kolmogorov, Brunk-Chung, and Marcinkiewicz-Zygmund type strong laws of large numbers for double sums in Rademacher type p (1 ≤ p ≤ 2) Banach spaces.

Sufficient and Necessary Conditions of Complete Convergence for Weighted Sums ofρ~-Mixing Random Variables

The equivalent conditions of complete convergence are established for weighted sums ofρ~-mixing random variables with different distributions. Our results extend and improve the Baum and Katz complete convergence theorem. As an application, the Marcinkiewicz-Zygmund type strong law of large numbers for sequence ofρ~-mixing random variables is obtained.

Complete convergence for weighted sums of arrays of rowwise ρ ˜ -mixing random variables

Let { X n i , i ≥ 1 , n ≥ 1 } Open image in new window be an array of rowwise ρ ˜ Open image in new window-mixing random variables. Some sufficient conditions for complete convergence for weighted sums of arrays of rowwise ρ ˜ Open image in new window-mixing random variables are presented without assumptions of identical distribution. As applications, the Baum and Katz type result and the Marcinkiewicz-Zygmund type strong law of large numbers for sequences of ρ ˜ Open image in new window-mixing random variables are obtained.

Strong convergence results for weighted sums of ρ ˜ -mixing random variables

In this paper, the authors study the strong convergence for weighted sums of ˜ ρ-mixing random variables without assumption of identical distribution. As an application, the Marcinkiewicz-Zygmund type strong law of large numbers for weighted sums of ˜ ρ-mixing random variables is obtained. MSC: 60F15