Tien-Chung Hu

On complete convergence for arrays of rowwise independent random elements in Banach spaces

We extend and generalize some recent results on complete convergence (cf. Hu, Moricz, and Taylor [14], Gut [11], Wang, Bhaskara Rao, and Yang [26], Kuczmaszewska and Szynal [17], and Sung [23]) for arrays of rowwise independent Banach space valued random elements. In the main result, no assumptions are made concerning the existence of expected values or absolute moments of the random elements and no assumptions are made concerning the geometry of the underlying Banach space. Some well-known results from the literature are obtained easily as corollaries. The corresponding convergence rates are also established

On the strong law for arrays and for the bootstrap mean and variance

Chung type strong laws of large numbers are obtained for arrays of rowwise independent random variables under various moment conditions. An interesting application of these results is the consistency of the bootstrap mean and variance.

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

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

Variable location and scale kernel density estimation

Variable (bandwidth) kernel density estimation (Abramson (1982,Ann. Statist.,10, 1217–1223)) and a kernel estimator with varying locations (Samiuddin and El-Sayyad (1990,Biometrika,77, 865–874)) are complementary ideas which essentially both afford bias of orderh4 as the overall smoothing parameterh → 0, sufficient differentiability of the density permitting. These ideas are put in a more general framework in this paper. This enables us to describe a variety of ways in which scale and location variation may be extended and/or combined to good theoretical effect. This particularly includes extending the basic ideas to provide new kernel estimators with bias of orderh6. Technical difficulties associated with potentially overly large variations are fully accounted for in our theory.

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].

ON THE RATE OF CONVERGENCE IN THE STRONG LAW OF LARGE NUMBERS FOR ARRAYS

For sequences of independent and identically distributed random variables it is well known that the existence of the second moment implies the law of the iterated logarithm. We show that the of the iterated logarithm does not extend to arrays of independent and identically distributed random variables and we develop an analogous rate result for such arrays under finitefourth moments

Strong laws of large numbers for arrays of rowwise independent random elements

Let {Xnk} be an array of rowwise independent random elements in a separable Banach space of type p

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.