Conditional limit theorems for conditionally negatively associated random variables

From the ordinary notion of negative association for a sequence of random variables, a new concept called conditional negative association is introduced. The relation between negative association and conditional negative association is answered, that is, the negative association does not imply the conditional negative association, and vice versa. The basic properties of conditional negative association are developed, which extend the corresponding ones under the non-conditioning setup. By means of these properties, some Rosenthal type inequalities for maximum partial sums of such sequences of random variables are derived, which extend the corresponding results for negatively associated random variables. As applications of these inequalities, some conditional mean convergence theorems, conditionally complete convergence results and a conditional central limit theorem stated in terms of conditional characteristic functions are established. In addition, some lemmas in the context are of independent interest.     

Conditional versions of limit theorems for conditionally associated random variables

Relation between association and conditional association is answered, several examples show that the association of random variables does not imply the conditional association, and vice versa. Several fundamental properties of conditional associated random variables are developed, which extend the corresponding ones under the non-conditioning setup. By means of these properties, some conditional Hájek-Rényi type inequalities, a conditional strong law of large numbers and a conditional central limit theorem stated in terms of conditional characteristic functions are established, which are conditional versions of the earlier results for associated random variables, respectively. In addition, some lemmas in the context are of independent interest.     

Some new limit theorems for vector space- and group-valued random variables

It is shown that weakly convergent sums (products) of normalized i.i.d. random variables with values in a finite-dimensional vector space or in a group are mixing in the sense of A. Rényi, and limit theorems for random sums with nonindependent indices are obtained. A new version of H. Robbins' limit theorem for random sums is presented. Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló Hungary, 1997, Part III.     

On central limit theorems for shrunken random variables

We discuss Central Limit Theorems and absence of limiting distributions for shrunken random variables.

     

Central limit theorems for generalized set-valued random variables

We give central limit theorems for generalized set-valued random variables whose level sets are compact both in or in a Banach space under milder conditions than those obtained recently by the latter two authors.     

Almost sure limit theorems for random sums of multiindex random variables

In this paper, we obtain an almost sure functional limit theorem for random sums of multiindex random variables.     

Central limit theorem for associated random variables

In this paper, we investigate an functional central limit theorem for a nonstatioaryd-parameter array of associated random variables applying the criterion of the tightness condition in Bickel and Wichura[1971]. Our results imply an extension to the nonstatioary case of invariance principle of Burton and Kim(1988) and analogous results for thed-dimensional associated random measure. These results are also applied to show a new functional central limit theorem for Poisson cluster random variables.     

One-sided limit theorems for sums of multidimensional indexed random variables

Let {X,X _n,nZ ₊ ^d } be a sequence of independent and identically distributed random variables and {a _n ,n Z ₊ ^d } be a sequence of constants. We examine the almost sure limiting behavior of weighted partial sums of the form _|n|N a _n X _n . Suppose further that eitherEX=0 orE|X|=. In most situations these normalized partial sums fail to have a limit, no matter which normalizing sequence we choose. Thus, the investigation lends itself to the study of the limit inferior and limit superior of these sequences. On the way to proving results of this type we first establish several weak laws. These weak laws prove to be of great value in establishing generalized laws of the iterated logarithm.     

Almost sure versions of limit theorems for random sums of multiindex random variables

In this paper we obtain an almost sure version of a limit theorem for random sums of multiindex random variables that belong to the domain of attraction of a p-stable law.     

Central limit theorems for C(S)-valued random variables

Let C(S) be the space of real-valued continuous functions on a compact metric space S. Let {X_n, n ? 1} be a sequence of independent identically distributed C(S)-valued random variables with mean zero and sup_t?sE[X₁²(t)] = 1. We show that the measures induced by ($\text{X}{}_1\text{+ ··· + X}{}_{\mathrm{n}}\text{) n}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}$ converge weakly to a Gaussian measure on C(S) under different conditions on X₁, one of which consolidates and extends results of Strassen and Dudley, Giné, and Dudley. Our method of proof is different from the methods employed by these authors.     

Exponential inequalities for N-demimartingales and negatively associated random variables

The class of N-demimartingales generalizes in a natural way the concept of negative association and includes as special cases martingales with respect to the natural choice of σ-algebras. For this class of random variables, a number of maximal and other inequalities were obtained by [Christofides, T.C., 2003. Maximal inequalities for N-demimartingales. Archives of Inequalities and Applications 50, 397–408] and [Prakasa Rao, B.L.S., 2004. On some inequalities for N-demimartingales. J. Indian Soc. Agricultural Statist. 57, 208–216; Prakasa Rao, B.L.S., 2007. On some maximal inequalities for demisubmartingales and N-demisupermartingales. J. Inequal. Pure Appl. Math. 8, 17]. In this paper we prove Azuma’s inequality for N-demimartingales and as a corollary we obtain an exponential inequality for negatively associated random variables.     

Conditional limit theorems for conditionally linearly negative quadrant dependent random variables

From the ordinary notion of linearly negative quadrant dependence for a sequence of random variables, a new concept called conditionally linearly negative quadrant dependence is introduced. The relation between the two kinds of dependence is answered by examples, that is, the linearly negative quadrant dependence does not imply the conditionally linearly negative quadrant dependence, and vice versa. The fundamental properties of conditionally linearly negative quadrant dependence are developed, which extend the corresponding ones under the non-conditioning setup. By means of these properties, some conditional exponential inequalities, conditionally complete convergence results and a conditional central limit theorem stated in terms of conditional characteristic functions are established.     

矩条件下的任意随机变量序列的一类强极限定理

本文利用随机变量的截尾方法和条件三级数定理,研究了任意随机变量序列在矩条件下的一类强极限定理,改进了与此相应的一些结果的条件.