A strong law of large numbers for independent random variables under non-additive probabilities

Abstract

Under non‐additive probabilities, cluster points of the empirical average have been proved to quasi-surely fall into the interval constructed by either the lower and upper expectations or the lower and upper Choquet expectations. In this paper, based on the initiated notion of independence, we obtain a different Marcinkiewicz-Zygmund type strong law of large numbers. Then the Kolmogorov type strong law of large numbers can be derived from it directly, stating that the closed interval between the lower and upper expectations is the smallest one that covers cluster points of the empirical average quasi-surely.     

Strong Law of Large Numbers for Countable Asymptotic Circular Markov Chains

In this article, we introduce the notion of a countable asymptotic circular Markov chain, and prove a strong law of large numbers: as a corollary, we generalize a well-known version of the strong law of large numbers for nonhomogeneous Markov chains, and prove the Shannon-McMillan-Breiman theorem in this context, extending the result for the finite case.     

Strong law of large numbers of the delayed sums for Markov Chains indexed by a Cayley tree

Abstract

In this paper, we will study the strong law of large numbers of the delayed sums for Markov chains indexed by a Cayley tree with countable state spaces. Firstly, we prove a strong limit theorem for the delayed sums of the bivariate functions for Markov chains indexed by a Cayley tree. Secondly, the strong law of large numbers for the frequencies of occurrence of states of the delayed sums is obtained. As a corollary, we obtain the strong law of large numbers for the frequencies of occurrence of states for countable Markov chains indexed by a Cayley tree.     

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.     

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 law of large numbers for generalized sample relative entropy of non homogeneous Markov chains

In this paper, we study the strong law of large numbers for the generalized sample relative entropy of non homogeneous Markov chains taking values from a finite state space. First, we introduce the definitions of generalized sample relative entropy and generalized sample relative entropy rate. Then, using a strong limit theorem for the delayed sums of the functions of two variables and a strong law of large numbers for non homogeneous Markov chains, we obtain the strong law of large numbers for the generalized sample relative entropy of non homogeneous Markov chains. As corollaries, we obtain some important results.     

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.     

Some extensions of the classical law of large numbers

Abstract

In the present article, we study the classic Bernoulli weak law of large numbers and Borel strong law of large numbers, which weaken the assumptions of some known results.     

Limit theorems for identically distributed martingale difference

Abstract

In this paper, we establish some general results for the strong law of large numbers and the complete convergence of martingale difference which include the well-known Marcinkiewicz–Zygmund strong law and Spitzer complete convergence.     

The generalizations of strong law of large numbers for asymptotic even–odd Markov chains indexed by a homogeneous tree

Yang et al. (Yang et al., J. Math. Anal. Appl., 410 (2014), 179–189.) have obtained the strong law of large numbers and asymptotic equipartition property for the asymptotic even–odd Markov chains indexed by a homogeneous tree. In this article, we are going to study the strong law of large numbers and the asymptotic equipartition property for a class of non homogeneous Markov chains indexed by a homogeneous tree which are the generalizations of above results. We also provide an example showing that our generalizations are not trivial.     

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.     

Strong laws of large numbers for sub-linear expectation without independence

In this paper, we investigate some strong laws of large numbers for sub-linear expectation without independence which generalize the classical ones. We give some strong laws of large numbers for sub-linear expectation on some moment conditions with respect to the partial sum and some conditions similar to Petrov’s. We can reduce the conclusion to a simple form when the the sequence of random variables is i.i.d. We also show a strong law of large numbers for sub-linear expectation with assumptions of quasi-surely.     

Asymptotic equipartition property for second-order circular Markov chains indexed by a two-rooted Cayley tree

In this paper, we introduce a model of a second-order circular Markov chain indexed by a two-rooted Cayley tree and establish two strong law of large numbers and the asymptotic equipartition property (AEP) for circular second-order finite Markov chains indexed by this homogeneous tree. In the proof, we apply a limit property for a sequence of multi-variable functions of a non homogeneous Markov chain indexed by such tree. As a corollary, we obtain the strong law of large numbers and AEP about the second-order finite homogeneous Markov chain indexed by the two-rooted homogeneous tree.     

ON THE STRONG LAW OF LARGE NUMBERS UNDER REARRANGEMENTS FOR SEQUENCES OF BLOCKWISE ORTHOGONAL RANDOM ELEMENTS IN BANACH SPACES

The condition of the strong law of large numbers is obtained for sequences of random elements in type p Banach spaces that are blockwise orthogonal. The current work extends a result of Chobanyan & Mandrekar (2000) [On Kolmogorov SLLN under rearrangements for orthogonal random variables in a B ‐space. J. Theoret. Probab. 13, 135–139.] Special cases of the main results are presented as corollaries, and illustrative examples are provided.     

A Strong Law of Large Numbers for Strongly Mixing Processes

We prove a strong law of large numbers for a class of strongly mixing processes. Our result rests on recent advances in understanding of concentration of measure. It is simple to apply and gives finite-sample (as opposed to asymptotic) bounds, with readily computable rate constants. In particular, this makes it suitable for analysis of inhomogeneous Markov processes. We demonstrate how it can be applied to establish an almost-sure convergence result for a class of models that includes as a special case a class of adaptive Markov chain Monte Carlo algorithms.     

Maximal inequalities and strong law of large numbers for sequences of m-asymptotically almost negatively associated random variables

In this paper, we establish some inequalities for maximum of partial sums of m-asymptotically almost negatively associated random variables. With the help of these inequalities we prove some strong law of large numbers.     

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.     

Laws of large numbers for bootstrappedU-statistics

For the bootstrapped mean, a strong law of large numbers is obtained under the assumption of finiteness of the rth moment, for some r>1, and a weak law of large numbers is obtained under the finiteness of the first moment. The results are then extended to bootstrapped U-statistics under parallel conditions. Stochastic convergence of the jackknifed estimator of the variance of a bootstrapped U-statistic is proved. The asymptotic normality of the bootstrapped pivot and the bias of the bootstrapped U-statistic are indicated.     

Exponential inequalities for sums of unbounded ϕ-mixing sequence and their applications

In the article, the exponential inequalities for sums of unbounded ?-mixing sequence are given, which generalize the corresponding one for independent and identically distributed random variables. As applications, the strong law of large numbers and strong growth rate for ?-mixing random variables are obtained.