 
Summary: University of Washington Math 523A Lecture 8
Lecturer: Eyal Lubetzky
Friday, April 24, 2009
1 Strong, weak, and veryweak martingales
Our goal will be to reach the following theorem, which we first state informally:
Theorem 1.1 (Kallenberg, Sztencel `91 ; Hayes `05). Let X = (Xn)n0 be a martingale in
Rd
with X0 = 0. Then
P( Xn a) 2e1
(a1)2
2n a > 0.
Notice the remarkable fact that the bound doesn't depend on the dimension d. We will
see that it is sufficient to project the process to 2 dimensions.
What do we mean by a martingale in Rd
? We make the following definitions generalizing
the notion of a martingale:
Definition 1.2 (Strong, weak, and veryweak martingales). Suppose X = (Xt)t0 is a
process in Rd
satisfying E Xt < for all t (where · denotes the Euclidean norm in Rd
).
