On Tail Probabilities for Martingales
TL;DR: In this paper, the Laplace transform of the crossing time of a martingale with uniformly bounded increments is shown to have the same distribution as the distribution of crossing times of Brownian motion, even in the tail.
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Abstract: Watch a martingale with uniformly bounded increments until it first crosses the horizontal line of height $a$. The sum of the conditional variances of the increments given the past, up to the crossing, is an intrinsic measure of the crossing time. Simple and fairly sharp upper and lower bounds are given for the Laplace transform of this crossing time, which show that the distribution is virtually the same as that for the crossing time of Brownian motion, even in the tail. The argument can be adapted to extend inequalities of Bernstein and Kolmogorov to the dependent case, proving the law of the iterated logarithm for martingales. The argument can also be adapted to prove Levy's central limit theorem for martingales. The results can be extended to martingales whose increments satisfy a growth condition.
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TL;DR: In this article, upper bounds for the probability that the sum S of n independent random variables exceeds its mean ES by a positive number nt are derived for certain sums of dependent random variables such as U statistics.
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Probability Inequalities for the Sum of Independent Random Variables
TL;DR: In this article, a number of inequalities which improve on existing upper limits to the probability distribution of the sum of independent random variables are presented, which are applicable when the number of component random variables is small and/or have different distributions.
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