We also obtain the convergence rate for the strong law of large numbers, which improves the corresponding ones of kim and kim. As an application, the complete convergence theorem for weighted sums of aana random variables is obtained. 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. Conditional expectation of negatively associated random variables. In addition some improvements of basic properties of negatively associated random variables are provided. 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. The weak convergence for functions of negatively associated.
In addition, we present some sufficient conditions to prove the strong law of large. We establish the marcinkiewiczzygmundtype strong laws of large numbers for certain class of multilinear ustatistics based on 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. A remark on complete convergence for arrays of rowwise. 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 broad list of interesting examples of determinantal point processes. For negatively associated random variables, various exponential inequalities have been stated in the last decade. The weak convergence for functions of negatively associated random variables1 lixin zhang zhejiang university, hangzhou, peoples republic of china email.
Request pdf summability methods and negatively associated random variables the paper studies convergence of sequences of negatively associated random variables under. Wittmann type strong laws of large numbers for blockwise. An invariance principle for negatively associated random variables zhengyan lin 1 chinese science bulletin volume 42, pages 359 364 1997 cite this article. Conditional expectation of negatively associated random. Some limit theorems for 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. 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. 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.
Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. The bounded case has been examined by christofides and hadjikyriakou 9, jabbari. Sufficient and necessary conditions of complete convergence. The inequality improves the corresponding result which was obtained in kim, t. 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. Negative association cmu school of computer science. Pdf strong laws for certain types of u statistics based on. 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. Pdf a remark on complete convergence for arrays of. An invariance principle for negatively associated random. The following committee members have found the thesis acceptable in form and content, and that the candidate demonstrated satisfactory knowledge of the subject material. Pdf exponential inequality for negatively associated random.
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\. An exponential inequality is established for identically distributed negatively associated random variables which have the finite laplace transforms. In the present paper, we are interested in the asymptotically almost negatively associated random variables. Dirichlet random variables are always negatively associated. Some deviation inequalities for sums of negatively associated. Complete convergence for maximal sums of negatively. Steins method for negatively associated random variables. On the complete convergence of sums of negatively associated.
Kolmogorovtype and marcinkiewicztype strong laws of large. 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. Large deviation principles and moderate deviation upper bounds for stationary mnegatively associated random variables are proved. Exponential inequality for negatively associated random variables article pdf available in statistical papers 502. Pdf on the strong law for asymptotically almost negatively. Complete convergence for asymptotically almost negatively. Necessary and sufficient conditions are given for the complete convergence of maximal sums of identically distributed negatively associated random variables. As an application of the main results, the marcinkiewiczzygmund type strong law of large numbers based on weighted sums of ana cases is obtained. Complete convergence for arrays of rowwise asymptotically.
For instance, wu and jiang obtained complete convergence for negatively associated sequences of random variables. Let be an array of rowwise asymptotically almost negatively associated random variables. Under some suitable conditions, the central limit theorem and the weak convergence for sums. Pdf a remark on complete convergence for arrays of rowwise. Limiting behaviour of moving average processes based on a sequence ofmixing and negatively associated random variables submitted by d. In another application it is shown that negatively correlated normal random variables are na. Almost sure central limit theorem for selfnormalized. Strong limit theorems for weighted sums of negatively.
Convergence for sums of asymptotically almost negatively. Some sufficient conditions for the strong law of large numbers of random variables are presented. 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. 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. An exponential inequality for negatively associated random.
The basic properties of negative association are derived. Rocky mountain journal of mathematics project euclid. Some deviation inequalities for sums of negatively associated random variables. The proof is based on a rosenthal type maximal inequality, a kolmogorov type exponential inequality and steins method. As a result, we extend some complete convergence and complete moment convergence theorems for independent random variables to the. Some deviation inequalities for sums of negatively. 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. In addition, the results of the paper generalize and improve earlier ones of chung am j math 69. Negative association of random variables with applications jstor. Some strong convergence theorems for asymptotically.
We give an exponential inequality for negatively associated random variables. On the almost sure convergence for a linear process. This paper proves that the law of the iterated logarithm holds for a stationary negatively associated sequence of random variables with finite variance. Some sufficient conditions for complete convergence for arrays of rowwise asymptotically almost negatively associated random variables are presented without assumptions of identical distribution. Negative association definition, properties, and applications. Nov 23, 2018 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. Tail behavior of negatively associated heavytailed sums. On complete convergence for arrays of rowwise negatively. Say that the measure p on bn is positively associated if. Nathakhun wiroonsri submitted on 6 oct 2017 v1, last revised 8 sep 2018 this version, v3.
This function is called a random variable or stochastic variable or more precisely a random function stochastic function. 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. On negatively associated random variables springerlink. 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. Large deviations and moderate deviations for mnegatively. Negatively associated random variables, in an oral examination held on april 19, 2017. Steins method for negatively associated random variables with applications to second order stationary random fields authors. Exponential inequality for negatively associated random. Despite substantial progress in the theory of negatively associated random variables, there are still many unsolved problems. Moment inequalities for mnegatively associated random. Wittmann type strong laws of large numbers for blockwise m. Random variables and probability distributions random variables suppose that to each point of a sample space we assign a number. On the almost sure convergence for a linear process generated. 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.
In this paper we investigate the validity of these results under more general. In this paper, we establish negative dependence in the examples mentioned above in. The conditions are expressed in terms of integrability of random variables. 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. Aana random variables, complete moment convergence, l. These theorems obtained extend and improve some earlier results. Two random variables x, y are called negatively correlated, if covx. A remark on complete convergence for arrays of rowwise negatively associated random variables. Large deviation principles and moderate deviation upper bounds for stationary m negatively. Our result improves those of kim and kim 10, nooghabi and azarnoosh 11, and xing et al. As it is shown by examples in joagdev and proschan 4, if we are given a. Some types of convergence for negatively dependent random.
Negative dependence via the fkg inequality math chalmers. Some strong laws of large numbers for weighted sums of. We then have a function defined on the sample space. The inequality improves the results of kim and kim 2007, nooghabi and azarnoosh 2009, and xing et al. 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. Strong and weak convergence for asymptotically almost. Recently, the work gw18 developed an l1 version of steins method adapted to sums of positively associated random variables with applications to statistical physics. A comparison theorem on moment inequalities between. Summability methods and negatively associated random. Results on the asymptotic tail probabilities of the quantities, and s n max 0. In addition, we present some sufficient conditions to prove the strong law of large numbers for. As an application, the marcinkiewicz strong law of large numbers for aana random variables is obtained.
Pdf strong laws for certain types of u statistics based. May 19, 2009 an exponential inequality is established for identically distributed negatively associated random variables which have the finite laplace transforms. Other na distributions are the a multinomial, b convolution of unlike multinomials, c multivariate hypergeometric, d dirichlet, and e dirichlet compound multinomial. On negative association of some finite point processes on general. 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. Complete convergence and complete moment convergence for.
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