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Conditional Generalizations of Strong Laws Which Conclude the Partial Sums Converge Almost Surely
T. P. Hill
The Annals of Probability
Vol. 10, No. 3 (Aug., 1982), pp. 828-830
Published by: Institute of Mathematical Statistics
Stable URL: http://www.jstor.org/stable/2243392
Page Count: 3
You can always find the topics here!Topics: Conditional convergence, Random variables, Perceptron convergence procedure, Partial sums, Martingales, Distributivity, Property law, Mathematical theorems, Mathematics
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Suppose that for every independent sequence of random variables satisfying some hypothesis condition H, it follows that the partial sums converge almost surely. Then it is shown that for every arbitrarily-dependent sequence of random variables, the partial sums converge almost surely on the event where the conditional distributions (given the past) satisfy precisely the same condition H. Thus many strong laws for independent sequences may be immediately generalized into conditional results for arbitrarily-dependent sequences.
The Annals of Probability © 1982 Institute of Mathematical Statistics