Open Access
Translator Disclaimer
February, 1975 On Tail Probabilities for Martingales
David A. Freedman
Ann. Probab. 3(1): 100-118 (February, 1975). DOI: 10.1214/aop/1176996452


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.


Download Citation

David A. Freedman. "On Tail Probabilities for Martingales." Ann. Probab. 3 (1) 100 - 118, February, 1975.


Published: February, 1975
First available in Project Euclid: 19 April 2007

zbMATH: 0313.60037
MathSciNet: MR380971
Digital Object Identifier: 10.1214/aop/1176996452

Primary: 60F10
Secondary: 60F05 , 60F15 , 60G40 , 60G45

Keywords: central limit theorem , crossing time , Law of the iterated logarithm , martingale , tail probability

Rights: Copyright © 1975 Institute of Mathematical Statistics


Vol.3 • No. 1 • February, 1975
Back to Top