Accession Number : AD0656485

Title :   SOME RESULTS ON ALMOST SURE AND COMPLETE CONVERGENCE IN THE INDEPENDENT AND MARTINGALE CASES,

Corporate Author : PURDUE UNIV LAFAYETTE IND DEPT OF STATISTICS

Personal Author(s) : Stout,William Fleming

Report Date : AUG 1967

Pagination or Media Count : 72

Abstract : Let (Omega,F,P) be a probability space, (D(subscript n), n = or > 1) be a sequence of independent random variables, a(subscript nk) be a matrix of real numbers, T(subscript nm) = Summation, k=1 to k=m, of a(subscript nk) D(subscript k), and T(subscript n) be the almost sure limit as m approaches infinity when it exists. T(subscript n) is said to converge completely to zero (15) if Summation, n=1 to n=infinity, of p((absolute value of T subscript n) > epsilon) < infinity for all epsilon > o. Various conditions are given for the complete or almost sure convergence of T(subscript n) to zero, extending or improving results given by others. In Chapter II, we extend to the martingale case a result of Chow concerning the complete convergence of T(subscript n) to zero where the D sub n's are generalized Gaussian. In Chapter III a number of almost sure convergence results are established in the martingale case. Chapter IV an extension of the Kolmogorov law of the iterated logarithm to the martingale case is made.

Descriptors :   (*STATISTICAL ANALYSIS, RANDOM VARIABLES), (*SEQUENCES(MATHEMATICS), CONVERGENCE), PROBABILITY, MATRICES(MATHEMATICS), SERIES(MATHEMATICS), THEOREMS, INEQUALITIES

Subject Categories : Statistics and Probability

Distribution Statement : APPROVED FOR PUBLIC RELEASE