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 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.     

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.     

Strong law of large numbers and complete convergence for non-identically distributed WOD random variables

In this paper, we establish the strong law of large numbers and complete convergence for non-identically distributed WOD random variables. We derive some new inequalities of Fuk–Nagaev type for the sums of non-identically distributed WD random variables. All these results further extend and refine previous ones.     

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.     

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.     

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].     

On exact strong laws of large numbers for ratios of random variables and their applications

Abstract

We study the almost sure convergence of weighted sums of ratios of independent random variables satisfying some general, mild conditions. The obtained results are applied to exact laws for order statistics. An exact law for independent random variables which are nonidentically distributed is also proved and applied to ratios of adjacent order statistics for a sample of uniformly distributed random variables.     

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.     

A law of large numbers for the scaled age distribution of linear birth-and-death processes

It is shown that certain measure-valued stochastic processes describing the age distribution of particles whose development is controlled by linear critical birth-and-death processes converge in distribution to a deterministic positive bounded measure.     

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.     

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 strong law of large numbers for non homogeneous M-bifurcating Markov chains indexed by a M-branch Cayley tree

This article is devoted to the strong law of large numbers and the entropy ergodic theorem for non homogeneous M-bifurcating Markov chains indexed by a M-branch Cayley tree, which generalizes the relevant results of tree-indexed nonhomogeneous bifurcating Markov chains. Meanwhile, our proof is quite different from the traditional method.     

A Note on the Convergence Rate of Strong Law of Large Numbers for Positively Associated Sequences

By the inequalities established in this article, we obtain the convergence rate of strong law of large numbers for positively associated sequences. The results derived extend and improve the corresponding ones in Vronskii (1999 Vronskii , M. A. ( 1999 ). Rate of convergence in the slln for associated sequences and fields . Theory Probab. Appl. 43 ( 3 ): 449 – 462 .[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.     

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 strong law of large numbers and the Shannon-McMillan theorem for the mth-order nonhomogeneous Markov chains indexed by an m rooted Cayley tree

ABSTRACT

In this article, we studied the strong law of large numbers(LLN) and Shannon-McMillan theorem for an mth-order nonhomogeneous Markov chain indexed by an m- rooted Cayley tree. This article generalized the relative results of level mth-order nonhomogeneous Markov chains indexed by an m- rooted Cayley tree.     

Convergence rates in the law of large numbers for END linear processes with random coefficients

Abstract

In this paper, we consider convergence rates in the Marcinkiewicz–Zygmund law of the large numbers for the END linear processes with random coefficients. We extend some results of Baum and Katz (1965 Baum, L. E., and M. Katz. 1965. Convergence rates in the law of large numbers. Transactions of the American Mathematical Society 120 (1):108–23. doi: 10.2307/1994170.[Crossref], [Web of Science ®] , [Google Scholar]) to the case of dependent linear processes with the random coefficients.