SOME RESULTS ON ALMOST SURE AND COMPLETE CONVERGENCE IN THE INDEPENDENT AND MARTINGALE CASES,
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.
Document Details
- Document Type
- Technical Report
- Publication Date
- Aug 01, 1967
- Accession Number
- AD0656485
Entities
People
- William Fleming Stout
Organizations
- Purdue University