The Annals of Probability

On Tail Probabilities for Martingales

David A. Freedman

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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.

Article information

Ann. Probab., Volume 3, Number 1 (1975), 100-118.

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60F10: Large deviations
Secondary: 60F15: Strong theorems 60F05: Central limit and other weak theorems 60G45 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

Martingale crossing time tail probability law of the iterated logarithm central limit theorem


Freedman, David A. On Tail Probabilities for Martingales. Ann. Probab. 3 (1975), no. 1, 100--118. doi:10.1214/aop/1176996452.

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