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]).     

Strong Limit Theorems for Pairwise NQD Random Variables

A number of strong laws of large numbers for sequences of pairwise negative quadrant dependent (NQD) random variables have been established by using the generalized three series theorem. In this article, we obtain a strong law of large numbers by using the truncation technique and method of subsequences instead of the generalized three series theorem. Our result generalizes and improves on the corresponding one in Li and Yang (2008 Li , R. , Yang , W. ( 2008 ). Strong convergence of pairwise NQD random sequences . J. Math. Anal. Appl. 344 : 741 – 747 .[Crossref], [Web of Science ®] , [Google Scholar]). We also obtain a complete convergence result for an array of rowwise pairwise NQD random variables.     

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.     

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.     

The Strong Consistency of M Estimator in a Linear Model for Negatively Dependent Random Samples

The strong consistency of M estimators of the regression parameters in linear models for negatively dependent random errors under some mild conditions is established, which is an essential improvement on the relevant results in the literature on the moment conditions and dependent errors. Especially, Theorems 1 and 2 of Wu (2006 Wu , Q. Y. ( 2006 ). Strong consistency of M estimator in linear model for negatively associated samples . J. Syst. Sci. Complex. 19 ( 4 ): 592 – 600 .[Crossref] , [Google Scholar]) are not only extended to the case of negatively dependent random errors, but also are improved essentially on the moment conditions.     

A Strong Limit Theorem for Weighted Sums of Negatively Dependent Random Variables

In this article, we establish a new complete convergence theorem for weighted sums of negatively dependent random variables. As corollaries, many results on the almost sure convergence and complete convergence for weighted sums of negatively dependent random variables are obtained. In particular, the results of Jing and Liang (2008 Jing, B.Y., Liang, H.Y. (2008). Strong limit theorems for weighted sums of negatively associated random variables. J. Theor. Probab. 21:890–909.[Crossref], [Web of Science ®] , [Google Scholar]), Sung (2012 Sung, S.H. (2012). Complete convergence for weighted sums of negatively dependent random variables. Stat. Pap. 53:73–82.[Crossref], [Web of Science ®] , [Google Scholar]), and Wu (2010) can be obtained.     

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.     

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.     

On convergence rate in the SLLN for maximums of moving-average sums of ALNQD random fields

In this article, we define a notion of asymptotically linear negatively quadrant dependence and establish the rate of complete convergence for maximums of moving-average sums of asymptotically linear negatively quadrant dependent random fields.     

New ASA Fellows—1967

This paper presents at an elementary level a unified presentation of concepts related to sufficiency and minimal sufficiency. Extensively discussed are techniques for showing in a particular statistical model that a given statistic is not sufficient or that a given sufficient statistic is not minimal. The applicability of these techniques is illustrated in three examples.     

Linear Combinations, Products and Ratios of t Random Variables

Summary: Results on linear combinations, products, and ratios of t random variables are reviewed. We believe that this review will serve as an important reference and encourage further research activities in the area.     

On the Product and Ratio of Gamma and Beta Random Variables

Summary: The distributions of the product XY and the ratio X/Y are derived when X and Y are gamma and beta random variables distributed independently of each other. Tabulations of the associated percentage points and illustrations of their practical use are also provided. * The authors would like to thank the referee and the editor for carefully reading the paper and for their help in improving the paper.     

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.     

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.     

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.     

On an improved Erdő-Rényi-type law for increments of partial sums

Let X₁, X₂,… be an independently and identically distributed sequence with ξX₁ = 0, ξ exp (tX₁ < ∞ (t ≧ 0) and partial sums S_n = X₁ + … + X_n. Consider the maximum increment D₁ (N, K) = max_{0≤n ≤ N - K} (S_{n + K} - S_n)of the sequence (S_n) in (0, N) over a time K = K_N, 1 ≦ K ≦ N. Under appropriate conditions on (K_N) it is shown that in the case K_N/log N → 0, but K_N/(log N)^1/2 → ∞, there exists a sequence (α_N) such that K_-1/2 D₁ (N, K) - α_N converges to 0 w. p. 1. This result provides a small increment analogue to the improved Erd?s-Rényi-type laws stated by Csörg? and Steinebach (1981).