Andrei Volodin

Complete Convergence for Weighted Sums and Arrays of Rowwise Extended Negatively Dependent Random Variables

In this article, we study the complete convergence for weighted sums of extended negatively dependent random variables and row sums of arrays of rowwise extended negatively dependent random variables. We apply two methods to prove the results: the first of is based on exponential bounds and second is based on the generalization of the classical moment inequality for extended negatively dependent random variables.

On the rate of complete convergence for weighted sums of arrays of Banach space valued random elements with application to moving average processes

We obtain complete convergence results for arrays of rowwise independent Banach space valued random elements. In the main result no assumptions are made concerning the geometry of the underlying Banach space. As corollaries we obtain a result on complete convergence in stable type p Banach spaces and on the complete convergence of moving average processes.

On the Strong Rates of Convergence for Arrays of Rowwise Negatively Dependent Random Variables

The strong convergence rate and complete convergence results for arrays of rowwise negatively dependent random variables are established. The results presented generalize the results of Chen et al. [1] and Sung et al. [2]. As applications, some well-known results on independent random variables can be easily extended to the case of negatively dependent random variables.

WEAK LAWS OF LARGE NUMBERS FOR ARRAYS UNDER A CONDITION OF UNIFORM INTEGRABILITY

For an array of dependent random variables satisfying a new notion of uniform integrability, weak laws of large numbers are obtained. Our results extend and sharpen the known results in the literature.

On the rate of complete convergence for weighted sums of arrays of banach space valued random elements

By applying a recent result of Hu et al. [Stochastic Anal. Appl., 17 (1999), pp. 963--992], we extend and generalize the complete convergence results of Pruitt [ J. Math. Mech., 15 (1966), pp. 769--776] and Rohatgi [Proc. Cambridge Philos. Soc., 69 (1971), pp. 305--307] to arrays of row-wise independent Banach space valued random elements. No assumptions are made concerning the geometry of the underlying Banach space. Illustrative examples are provided comparing the various results.

COMPLETE CONVERGENCE FOR ARRAYS OF ROWWISE INDEPENDENT RANDOM VARIABLES

Abstract. Under some conditions on an array of rowwise inde-pendent random variables, Hu et al.(1998) obtained a completeconvergence result for law of large numbers with rate fa n ;n ‚ 1 g which is bounded away from zero. We investigate the general situ-ation for rate fa n ;n ‚ 1 g under similar conditions. 1. IntroductionThe concept of complete convergence of a sequence of random vari-ables was introduced by Hsu and Robbins [5] as follows. A sequence fU n ;n ‚ 1 g of random variables converges completely to the constant µ ifX 1n =1 P ( jU n i µj > † ) 0. We refer to [3] for a survey on results on complete conver-gence related to strong laws.Recently, Hu et al. [6] and Hu and Volodin [8] proved the follow-ing complete convergence theorem for arrays of rowwise independentrandom variables.Theorem 1. Let fX ni ; 1 • i • k n ;n ‚ 1 g be an array of rowwiseindependent random variables and fa n ;n ‚ 1 g a sequence of positive Received April 19, 2002.2000 Mathematics Subject Classification: 60F15, 60G50, 60B12.Key words and phrases: Arrays, rowwise independence, sums of independent ran-dom variables, complete convergence, Rademacher type

ON THE COMPLETE CONVERGENCE FOR ARRAYS OF ROWWISE EXTENDED NEGATIVELY DEPENDENT RANDOM VARIABLES

Abstract. A general result for the complete convergence of arrays ofrowwise extended negatively dependent random variables is derived. Asits applications eight corollaries for complete convergence of weightedsums for arrays of rowwise extended negatively dependent random vari-ables are given, which extend the corresponding known results for inde-pendent case. 1. IntroductionThe concept of complete convergence of a sequence of random variables wasintroduced by Hsu and Robbins ([5]) as follows. A sequence {U n ,n≥ 1} ofrandom variables converges completely to the constant θifX ∞n=1 P{|U n −θ| >ǫ} 0.Moreover, they proved that the sequence of arithmetic means of independentidentically distribution (i.i.d.) random variables converges completely to theexpected value if the variance of the summands is finite. This result has beengeneralized and extended in several directions, see Gut ([3], [4]), Hu et al. ([7],[8]), Chen et al. ([2]), Sung ([14], [15], [17]), Zarei and Jabbari ([20]), Baek etal. ([1]). In particular, Sung ([14]) obtained the following two Theorems A andB.Theorem A. Let {X

ON COMPLETE CONVERGENCE FOR ARRAYS OF ROW-WISE NEGATIVELY ASSOCIATED RANDOM VARIABLES

We obtain a complete convergence result for arrays of row-wise negatively associated random variables which extends the results of Hu, Szynal, and Volodin [Statist. Probab. Lett., 38 (1998), pp. 27–31], Hu et al. [Commun. Korean Math. Soc., 18 (2003), pp. 375–383], and Sung, Volodin, and Hu [Statist. Probab. Lett., 71 (2005), pp. 303–311] by the methods developed by Kruglov, Volodin, and Hu [Statist. Probab. Lett., 76 (2006), pp. 1631–1640].

STRONG LIMIT THEOREMS FOR WEIGHTED SUMS OF NOD SEQUENCE AND EXPONENTIAL INEQUALITIES

Some properties for negatively orthant dependent sequence are discussed. Some strong limit results for the weighted sums are ob- tained, which generalize the corresponding results for independent se- quence and negatively associated sequence. At last, exponential inequal- ities for negatively orthant dependent sequence are presented.

Addendum to “A note on complete convergence for arrays”, Statist. Probab. Lett. 38 (1) (1998) 27–31 ☆

Abstract Under some conditions on an array of rowwise-independent random variables, Hu, Szynal and Volodin obtained a complete convergence result for law of large numbers. In this addendum we mention that the convergent rate of sequence {c n , n⩾1} must be bounded away from zero.