## The Annals of Probability

### On Tail Probabilities for Martingales

David A. Freedman

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176996452

Digital Object Identifier
doi:10.1214/aop/1176996452

Mathematical Reviews number (MathSciNet)
MR380971

Zentralblatt MATH identifier
0313.60037

JSTOR