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.     

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.     

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.     

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.     

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.     

A class of small deviation theorems for the random variables associated with mth-order asymptotic circular Markov chains

ABSTRACT

In this article, we study a class of small deviation theorems for the random variables associated with mth-order asymptotic circular Markov chains. First, the definition of mth-order asymptotic circular Markov chain is introduced, then by applying the known results of the limit theorem for mth-order non homogeneous Markov chain, the small deviation theorem on the frequencies of occurrence of states for mth-order asymptotic circular Markov chains is established. Next, the strong law of large numbers and asymptotic equipartition property for this Markov chains are obtained. Finally, some results of mth-order nonhomogeneous Markov chains are given.     

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.     

Conditional entropy,entropy density,and strong law of large numbers for generalized controlled tree-indexed Markov chains

In this paper, we first introduces a tree model without degree boundedness restriction namely generalized controlled tree T, which is an extension of some known tree models, such as homogeneous tree model, uniformly bounded degree tree model, controlled tree model, etc. Then some limit properties including strong law of large numbers for generalized controlled tree-indexed non homogeneous Markov chain are obtained. Finally, we establish some entropy density properties, monotonicity of conditional entropy, and entropy properties for generalized controlled tree-indexed Markov chains.     

Some limit theorems of delayed sums for rowwise conditionally independent stochastic arrays

This paper is concerned with the asymptotic property of delayed sums for rowwise conditionally independent stochastic arrays. The main technique of the proofing is to construct non negative random variables with one parameter and to apply the Borel–Cantelli lemma to obtaining almost everywhere convergence. The relevant results for non homogeneous Markov chains indexed by a tree are extended.     

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.     

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

Abstract

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.     

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.     

The definition of tree-indexed Markov chains in random environment and their existence

This paper gives the definition of tree-indexed Markov chains in random environment with discrete state space, and then studies some equivalent theorems of tree-indexed Markov chains in random environment. Finally, we give the equivalence on tree-indexed Markov chains in Markov environment and double Markov chains indexed by a tree.     

Strong convergence properties for partial sums of asymptotically negatively associated random vectors in Hilbert spaces

Abstract

In this paper, we investigate the almost sure convergence for partial sums of asymptotically negatively associated (ANA, for short) random vectors in Hilbert spaces. The Khintchine-Kolmogorov type convergence theorem, three series theorem and the Kolmogorov type strong law of large numbers for partial sums of ANA random vectors in Hilbert spaces are obtained. The results obtained in the paper generalize some corresponding ones for independent random vectors and negatively associated random vectors in Hilbert spaces.     

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.     

A version of the Kolmogrov–Feller weak law of large numbers for maximal weighted sums of random variables

Abstract

In this paper we establish Kolmogrov–Feller weak law of large numbers for maximal weighted sums of i.i.d. random variables.     

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.     

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 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.