Ronald F. Patterson

A strong law of large numbers for arrays of rowwise negatively dependent random variables

A strong law of large numbers for arrays of rowwise negatively dependent random variables is obtained which relaxes the usual assumption of rowwise independence. The moment conditions of the main result are similar to previous results, and the stochastic bounded condition also provides a relaxation of the usual distributional assumptions.

Chung type strong laws for arrays of random elements and bootstrapping

Let {Xnk } be be an array of rowwise independent random elements in a separable Banach space. Chung type strong laws of large numbers are obtained under various moment conditions on the random elements and geometric type p, 1≤p≤2, conditions on the Banach space. Comparisons with existing results for arrays of random elements are provided to illustrate the strength of these results. The results can be directly applied to show the asymptotic validity of the bootstrap mean and variance for random functions

WEAK LAWS OF LARGE NUMBERS FOR ARRAYS OF ROWWISE NEGATIVELY DEPENDENT RANDOM VARIABLES

Weak laws of large numbers for arrays of rowwise negatively dependent random variables are obtained in this paper. The more general hypothesis of negative dependence relaxes the usual assumption of independence. The moment conditions are similar to previous results, and the stochastic bounded condition also provides a generalization of the usual distributional assumptions.

Strong laws of large numbers for triangular arrays of exchangeable random variables

Let be an array of row-wise exchangeable random elements in a separable Banach space. Strong laws of large numbers are obtained for under certain moment conditions on the random variables and a condition relating to nonorthogonality. By using reverse martingale techniques, similar results are obtained for triangular arrays of random elements inseparable Banach spaces which are row-wise exchangeable

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

Let Xnk } be an array of rowwse conditionally independent random elements in a separable Banach space of type p, 1 _< p _< 2. Complete convergence of n-I/r Xnk to 0, 0 < r < p _< 2 is obtained k=l by using various conditions on the moments and conditional means. A Chung type strong law of large numbers is also obtained under suitable moment conditions on the conditional means.

Strong convergence' for u–statistics in arrays of.row–wise exchangeable random variables

Let be an array of row–wise exchangeable random variables. Strong convergence results are obtained for U–statistics, using suitable moment conditions and martingale techniques. Statistics considered for exemplification are the sample variance and Spearmans Rank Correlation Coefficient

Strong convergence for sums of randomly weighted, rowwise exchangeable random variables

Let be an array of rowwise exchangeable random elements in a separable Banach space. Let {An} and {an} be random variables where An is positive and an is a symmetric function of . Using reverse martingale techniques, strong convergence is obtained for the weighted sum , under certain moment conditions on me random elements and suitable conditions on the random weights

STRONG LAWS OF LARGE NUMBERS FOR ARRAYS OF ROW-WISE EXCHANGEABLE RANDOM ELEMENTS

Let {Xnk,1≤k≤n,n≤1} be a triangular array of row-wise exchangeable random elements in a separable Banach space. The almost sure convergence of n−1/p∑k=1nXnk,1≤p<2, is obtained under varying moment and distribution conditions on the random elements. In particular, strong laws of large numbers follow for triangular arrays of random elements in(Rademacher) type p separable Banach spaces. Consistency of the kernel density estimates can be obtained in this setting.