SOME RESULTS ON ALMOST SURE AND COMPLETE CONVERGENCE IN THE INDEPENDENT AND MARTINGALE CASES,
PURDUE UNIV LAFAYETTE IND DEPT OF STATISTICS
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Let Omega,F,P be a probability space, Dsubscript n, n or 1 be a sequence of independent random variables, asubscript nk be a matrix of real numbers, Tsubscript nm Summation, k1 to km, of asubscript nk Dsubscript k, and Tsubscript n be the almost sure limit as m approaches infinity when it exists. Tsubscript n is said to converge completely to zero 15 if Summation, n1 to ninfinity, of pabsolute value of T subscript n epsilon infinity for all epsilon o. Various conditions are given for the complete or almost sure convergence of Tsubscript 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 Tsubscript n to zero where the D sub ns 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.
- Statistics and Probability