In another application it is shown that negatively correlated normal random variables are na. Mnegatively associated random variables, which generalizes the classical one of negatively associated random variables and includes mdependent sequences as its particular case, are introduced and studied. This function is called a random variable or stochastic variable or more precisely a random function stochastic function. For negatively associated random variables, various exponential inequalities have been stated in the last decade. In this paper, we study the complete convergence and complete moment convergence for negatively associated sequences of random variables with \\mathbbex0\, \\mathbbe\exp\ln\alphax 1\. In addition, the results of the paper generalize and improve earlier ones of chung am j math 69. As an application, the complete convergence theorem for weighted sums of aana random variables is obtained. Two random variables x, y are called negatively correlated, if covx. Some deviation inequalities for sums of negatively associated random variables. Some deviation inequalities for sums of negatively associated. The inequality improves the corresponding result which was obtained in kim, t. On negative association of some finite point processes on general. On the almost sure convergence for a linear process. Negative association of random variables with applications jstor.
Complete convergence for asymptotically almost negatively. Pdf on the strong law for asymptotically almost negatively. Random variables and probability distributions random variables suppose that to each point of a sample space we assign a number. Apr 18, 2012 in this note we show how chebyshevs other inequality can be applied to construct negatively associated random variables and to lead to a simplification of proofs for some known results on such random variables. Kolmogorovtype and marcinkiewicztype strong laws of large. As an application of the main results, the marcinkiewiczzygmund type strong law of large numbers based on weighted sums of ana cases is obtained. We establish the marcinkiewiczzygmundtype strong laws of large numbers for certain class of multilinear ustatistics based on negatively associated random variables. Recently, the work gw18 developed an l1 version of steins method adapted to sums of positively associated random variables with applications to statistical physics. As an application of the main results, the marcinkiewiczzygmund type strong law of large numbers based on weighted sums of ana cases is. In addition, the fellertype weak law of large number for sequences of aana.
The following committee members have found the thesis acceptable in form and content, and that the candidate demonstrated satisfactory knowledge of the subject material. Some limit theorems for negatively associated random variables. In this paper we investigate the validity of these results under more general. Pdf strong laws for certain types of u statistics based on. Large deviations and moderate deviations for mnegatively. The strong law of large numbers for sequences of asymptotically almost negatively associated aana, in short random variables is obtained, which generalizes and improves the corresponding one of bai and cheng 2000 for independent and identically distributed random variables to the case of aana random variables. An exponential inequality is established for identically distributed negatively associated random variables which have the finite laplace transforms. Summability methods and negatively associated random. The conditions are expressed in terms of integrability of random variables. Limiting behaviour of moving average processes based on a sequence ofmixing and negatively associated random variables submitted by d. Conditional expectation of negatively associated random. Complete convergence for arrays of rowwise asymptotically. Some strong laws of large numbers for weighted sums of.
Negative association cmu school of computer science. An invariance principle for negatively associated random. Our result improves those of kim and kim 10, nooghabi and azarnoosh 11, and xing et al. The weak convergence for functions of negatively associated. For instance, wu and jiang obtained complete convergence for negatively associated sequences of random variables. Complete convergence and complete moment convergence for. Wittmann type strong laws of large numbers for blockwise. Pdf some deviation inequalities for sums of negatively. Under some suitable conditions, the central limit theorem and the weak convergence for sums. Pdf a remark on complete convergence for arrays of rowwise. Conditional expectation of negatively associated random variables.
In this investigation, some sufficient and necessary conditions of the complete convergence for weighted sums of asymptotically negatively associated ana, in short random variables are presented without the assumption of identical distribution. We also obtain the convergence rate for the strong law of large numbers, which improves the corresponding ones of kim and kim. Some types of convergence for negatively dependent random. Necessary and sufficient conditions are given for the complete convergence of maximal sums of identically distributed negatively associated random variables. Some strong convergence theorems for asymptotically. The weak convergence for functions of negatively associated random variables1 lixin zhang zhejiang university, hangzhou, peoples republic of china email. In this paper, we establish an exponential inequality for identically distributed negatively associated random variables by using truncation method not using a block decomposition of the sums. Random variables, xi, xk are said to be negatively associated na if for every pair of disjoint. Large deviation principles and moderate deviation upper bounds for stationary mnegatively associated random variables are proved. In this article, the strong law of large numbers for weighted sums of asymptotically almost negatively associated aana, in short random variables is obtained. A remark on complete convergence for arrays of rowwise.
On complete convergence of weighted sums for arrays of rowwise asymptotically almost negatively associated random variables wang, xuejun, hu, shuhe, yang, wenzhi, and wang, xinghui, abstract and applied. Aana random variables, complete moment convergence, l. Negative dependence via the fkg inequality math chalmers. Pdf a remark on complete convergence for arrays of. Convergence for sums of asymptotically almost negatively. On the complete convergence of sums of negatively associated. Rocky mountain journal of mathematics project euclid. As a result, we extend some complete convergence and complete moment convergence theorems for independent random variables to. In the present paper, we are interested in the asymptotically almost negatively associated random variables.
An exponential inequality for negatively associated random. In addition some improvements of basic properties of negatively associated random variables are provided. Results on the asymptotic tail probabilities of the quantities, and s n max 0. In this paper, based on the initiation of the notion of negatively associated random variables under nonlinear probability, a strong limit theorem for weighted sums of random variables within the same frame is achieved without assumptions of independence and identical distribution, from which the marcinkiewichzygmund type and kolmogorov type strong laws of large numbers are derived. On the strong law for asymptotically almost negatively associated random variables article pdf available in rocky mountain journal of mathematics 343 september 2004 with 55 reads. M negatively associated random variables, which generalizes the classical one of negatively associated random variables and includes mdependent sequences as its particular case, are introduced and studied. The bounded case has been examined by christofides and hadjikyriakou 9, jabbari. The law of the iterated logarithm for negatively associated. As it is shown by examples in joagdev and proschan 4, if we are given a. In addition, we present some sufficient conditions to prove the strong law of large. The proof is based on a rosenthal type maximal inequality, a kolmogorov type exponential inequality and steins method. Say that the measure p on bn is positively associated if. Proofs are based on new maximal inequalities for sums of bounded negatively associated random variables.
A broad list of interesting examples of determinantal point processes. For unbounded negatively associated random variables, kim and kim 22, nooghabi and azarnoosh 28, sung 34, xing 40, xing and yang 42 and xing et al. On the almost sure convergence for a linear process generated. Steins method for negatively associated random variables. The basic properties of negative association are derived. Strong limit theorems for weighted sums of negatively. In this note we show how chebyshevs other inequality can be applied to construct negatively associated random variables and to lead to a simplification of proofs for some known results on such random variables. Other na distributions are the a multinomial, b convolution of unlike multinomials, c multivariate hypergeometric, d dirichlet, and e dirichlet compound multinomial. Wittmann type strong laws of large numbers for blockwise m. Request pdf summability methods and negatively associated random variables the paper studies convergence of sequences of negatively associated random variables under.
Complete convergence for maximal sums of negatively. Exponential inequality for negatively associated random variables article pdf available in statistical papers 502. As an application, the marcinkiewicz strong law of large numbers for aana random variables is obtained. Tail behavior of negatively associated heavytailed sums.
Some deviation inequalities for sums of negatively. Dirichlet random variables are always negatively associated. Pdf strong laws for certain types of u statistics based. Steins method for negatively associated random variables with applications to second order stationary random fields authors. Negative association definition, properties, and applications. Exponential inequality for negatively associated random. Let be an array of rowwise asymptotically almost negatively associated random variables. In addition, we present some sufficient conditions to prove the strong law of large numbers for. An invariance principle for negatively associated random variables zhengyan lin 1 chinese science bulletin volume 42, pages 359 364 1997 cite this article.
Negatively associated random variables, in an oral examination held on april 19, 2017. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. Moment inequalities for mnegatively associated random. Despite substantial progress in the theory of negatively associated random variables, there are still many unsolved problems. Nathakhun wiroonsri submitted on 6 oct 2017 v1, last revised 8 sep 2018 this version, v3. A remark on complete convergence for arrays of rowwise negatively associated random variables. In this paper, we establish negative dependence in the examples mentioned above in. We then have a function defined on the sample space. Sufficient and necessary conditions of complete convergence. Some sufficient conditions for complete convergence for arrays of rowwise asymptotically almost negatively associated random variables are presented without assumptions of identical distribution. In section 2 we will study the strong law of large numbers for negatively associated random variables in a hilbert space and in section 3 we derive the strong law of large numbers for a strictly stationary linear process generated by negatively associated random variables in a hilbert space by applying this result. Large deviation principles and moderate deviation upper bounds for stationary m negatively. A survey on negatively associated random variables can be found in the recent monograph by bulinski and shashkin 1. In particular, the comparison theorem on moment inequalities between negatively associated and independent random variables extends the hoeffding inequality on the probability bounds for the sum of a random sample without replacement from a finite population.
These theorems obtained extend and improve some earlier results. As a result, we extend some complete convergence and complete moment convergence theorems for independent random variables to the. Applications to limiting distributions of estimators of var s n are also discussed. Almost sure central limit theorem for selfnormalized. We give an exponential inequality for negatively associated random variables. On complete convergence for arrays of rowwise negatively. May 19, 2009 an exponential inequality is established for identically distributed negatively associated random variables which have the finite laplace transforms. This paper proves that the law of the iterated logarithm holds for a stationary negatively associated sequence of random variables with finite variance.
In this paper, based on the initiation of the notion of negatively associated random variables under nonlinear probability, a strong limit theorem for weighted sums of random variables within the same frame is achieved without assumptions of independence and identical distribution, from which the marcinkiewichzygmund type and kolmogorov type strong laws of large. Pdf exponential inequality for negatively associated random. On complete convergence of weighted sums for arrays of. On negatively associated random variables springerlink. In the paper, we study the strong law of large numbers for general weighted sums of asymptotically almost negatively associated random variables aana, in short with nonidentical distribution.
Strong and weak convergence for asymptotically almost. The inequality improves the results of kim and kim 2007, nooghabi and azarnoosh 2009, and xing et al. As applications, some wellknown results on independent random variables can be easily extended to the case of negatively associated random variables. Request pdf summability methods and negatively associated random variables the paper studies convergence of sequences of negatively associated random variables under various summability methods. This paper extends results on complete convergence in the law of large numbers for subsequences to the case of negatively associated nonidentically distributed random variables.
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