Weak and strong laws of large numbers for sub-linear expectation

Abstract

In this paper, we derive a new form of weak laws of large numbers for sub-linear expectation and establish the equivalence relation among this new form and the other two forms of weak laws of large numbers for sub-linear expectation. Moreover, we obtain the strong laws of large numbers for sub-linear expectation under a general moment condition by applying our new weak laws of large numbers.     

Complete convergence for arrays of row-wise ND random variables under sub-linear expectations

In this paper, complete convergence for arrays of row-wise ND random variables under sub-linear expectations is studied. As applications, the complete convergence theorems of weighted sums for negatively dependent random variables have been generalized to the sub-linear expectation space context. We extend some complete convergence theorems from the traditional probability space to the sub-linear expectation space and our results generalize corresponding results obtained by Ko.     

The strong laws of large numbers for countable non homogeneous hidden Markov models

In this article, we are going to study the strong laws of large numbers for countable non homogeneous hidden Markov models. First, we introduce the notion of countable non homogeneous hidden Markov models. Then, we obtain some properties for those Markov models. Finally, we establish two strong laws of large numbers for countable non homogeneous hidden Markov models. As corollaries, we obtain some known results of strong laws of large numbers for finite non homogeneous Markov chains.     

Complete convergence and the strong laws of large numbers for pairwise NQD random variables

The authors study the strong convergence for sequences of pairwise negatively quadrant dependent (NQD) random variables under some wide conditions, and present some new theorems on the complete convergence and the strong laws of large numbers. The obtained results extend and improve some theorems in existing literature.     

The laws of large numbers for Pareto-type random variables with infinite means

In this paper, we consider the laws of large numbers for NSD random variables satisfying Pareto-type distributions with infinite means. Based on the Pareto-Zipf distributions, some weak laws of large numbers for weighted sums of NSD random variables are obtained. Meanwhile, we show that a weak law for Pareto-Zipf distributions cannot be extended to a strong law. Furthermore, based on the two tailed Pareto distribution, a strong law of large numbers for weighed NSD random variables is presented. Our results extend the corresponding earlier ones.     

Marcinkiewicz–Zygmund and ordinary strong laws for empirical distribution functions and plug-in estimators

Both Marcinkiewicz–Zygmund strong laws of large numbers (MZ-SLLNs) and ordinary strong laws of large numbers (SLLNs) for plug-in estimators of general statistical functionals are derived. It is used that if a statistical functional is ‘sufficiently regular’, then an (MZ-)SLLN for the estimator of the unknown distribution function yields an (MZ-)SLLN for the corresponding plug-in estimator. It is in particular shown that many L-, V- and risk functionals are ‘sufficiently regular’ and that known results on the strong convergence of the empirical process of α-mixing random variables can be improved. The presented approach does not only cover some known results but also provides some new strong laws for plug-in estimators of particular statistical functionals.     

Strong laws of large numbers for dependent heterogeneous processes: a synthesis of recent and new results

This paper surveys recent developments in the strong law of large numbers for dependent heterogeneous processes. We prove a generalised version of a recent strong law for Lz-mixingales, and also a new strong law for Lpmixingales. These results greatly relax the dependence and heterogeneity conditions relative to those currently cited, and introduce explicit trade-offs between dependence and heterogeneity. The results are applied to proving strong laws for near-epoch dependent functions of mixing processes. We contrast several methods for obtaining these results, including mapping directly to the mixingale properties, and applying a truncation argument.     

Strong laws of large numbers for dependent heterogeneous processes: a synthesis of recent and new results

This paper surveys recent developments in the strong law of large numbers for dependent heterogeneous processes. We prove a generalised version of a recent strong law for Lz-mixingales, and also a new strong law for Lpmixingales. These results greatly relax the dependence and heterogeneity conditions relative to those currently cited, and introduce explicit trade-offs between dependence and heterogeneity. The results are applied to proving strong laws for near-epoch dependent functions of mixing processes. We contrast several methods for obtaining these results, including mapping directly to the mixingale properties, and applying a truncation argument.     

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 integral convergence for arrays of row-wise extended independent random variables under Sub-linear expectations

Abstract

In this paper, we establish the complete convergence and complete integral convergence for arrays of row-wise extended independent random variables under sub-linear expectation space with some conditions. At the same time we extend some complete convergence and complete integral convergence theorems from the classical probability space to the sub-linear expectation space. The results generalize corresponding results obtained by Wu et al. (2017 Zhang, L. X. 2016b. Exponential inequalities under the sub-linear expectations with applications to laws of the iterated logarithm. Science China Mathematics 59 (12):2503–26. doi: 10.1007/s11425-016-0079-1.[Crossref], [Web of Science ®] , [Google Scholar]).     

Weak and strong laws of large numbers for coherent lower previsions

We prove weak and strong laws of large numbers for coherent lower previsions, where the lower prevision of a random variable is given a behavioural interpretation as a subject's supremum acceptable price for buying it. Our laws are a consequence of the rationality criterion of coherence, and they can be proven under assumptions that are surprisingly weak when compared to the standard formulation of the laws in more classical approaches to probability theory.     

On the formulation of uniform laws of large numbers: a truncation approach

The paper develops a general framework for the formulation of generic uniform laws of large numbers. In particular, we introduce a basic generic uniform law of large numbers that contains recent uniform laws of large numbers by Andrews [2] and Hoadley [9] as special cases. We also develop a truncation approach that makes it possible to obtain uniform laws of large numbers for the functions under consideration from uniform laws of large numbers for truncated versions of those functions. The point of the truncation approach is that uniform laws of large numbers for the truncated versions are typically easier to obtain. By combining the basic uniform law of large numbers and the truncation approach we also derive generalizations of recent uniform laws of large numbers introduced in Pötscher and Prucha [15, 16].     

The strong laws of large numbers for weighted sums of extended negatively dependent random variables

In this paper, the strong laws of large numbers for maximum value of weighted sums of extended negatively dependent random variables are obtained, which improve and extend the corresponding ones for independent random variables and some dependent random variables.     

Hajek–Renyi-type inequality and strong law of large numbers for END sequences

In this paper, we get the Hajek–Renyi-type inequality under 0 < q ? 2 for a sequence of extended negatively dependent (END) random variables with concrete coefficients, which generalizes and extends the general Hajek–Renyi-type inequality. In addition, we obtain some new results of the strong laws of large numbers and strong growth rate for END sequences.     

A strong law of large number for negatively dependent and non identical distributed random variables in the framework of sublinear expectation

Abstract

In this article, in the framework of sublinear expectation initiated by Peng, we derive a strong law of large numbers (SLLN) for negatively dependent and non identical distributed random variables. This result includes and extends some existing results. Furthermore, we give two examples of our result for applications.     

Strong laws of large numbers for the mth-order asymptotic odd–even Markov chains indexed by an m-rooted Cayley tree

In this article, we will study the strong laws of large numbers and asymptotic equipartition property (AEP) for mth-order asymptotic odd–even Markov chains indexed by an m-rooted Cayley tree. First, the definition of mth-order asymptotic odd–even Markov chains indexed by an m-rooted Cayley tree is introduced, then the strong limit theorem for this Markov chains is established. Next, the strong laws of large numbers for the frequencies of ordered couple of states for mth-order asymptotic odd–even Markov chains indexed by an m-rooted Cayley tree are obtained. Finally, we prove the AEP for this Markov chains.     

Basic structure of the asymptotic theory in dynamic nonlineaerco nometric models,part i: consistency and approximation concepts

This is the first of two papers that provide an expository discussion of the basic structure of the asymptotic theory of M-estimators in dynamic nonlinear models and a revlew of the literature. In this paper we discuss consistency,uniform laws of large numbers and develop a new framework for laws of large numbers for dependent and heterogeneous processes, encompassing the theory of stochastically stable as well as near epoch dependent processes. This framework results in simplified catalogues of sufficient conditions for consistency. The second paper, Potscher and Prucha (1990b), deals with asymptotic distribution theory.     

Basic structure of the asymptotic theory in dynamic nonlineaerco nometric models, part i: consistency and approximation concepts

This is the first of two papers that provide an expository discussion of the basic structure of the asymptotic theory of M-estimators in dynamic nonlinear models and a revlew of the literature. In this paper we discuss consistency,uniform laws of large numbers and develop a new framework for laws of large numbers for dependent and heterogeneous processes, encompassing the theory of stochastically stable as well as near epoch dependent processes. This framework results in simplified catalogues of sufficient conditions for consistency. The second paper, Potscher and Prucha (1990b), deals with asymptotic distribution theory.     

Convergence properties for weighted sums of NSD random variables

Abstract

Let {X_n, n ? 1} be a sequence of negatively superadditive dependent (NSD, in short) random variables and {b_ni, 1 ? i ? n, n ? 1} be an array of real numbers. In this article, we study the strong law of large numbers for the weighted sums ∑ⁿ_{i = 1}b_niX_i without identical distribution. We present some sufficient conditions to prove the strong law of large numbers. As an application, the Marcinkiewicz-Zygmund strong law of large numbers for NSD random variables is obtained. In addition, the complete convergence for the weighted sums of NSD random variables is established. Our results generalize and improve some corresponding ones for independent random variables and negatively associated random variables.