Randomized Algorithms 2013 A Lecture 5 – Martingales , Closest Pairs , Hash Tables , Existential Proofs and Codes ∗

The question is whether these two requirements are equivalent and whether we can get the nice concentration for both of them. They are not equivalent as the following example shows. Consider a random walk on the integers with a reflecting wall at 0 where the first step chooses whether the walk is in the non negative or non positive integers. That is X0 = 0, the first step is random in {−1, 1} and all subsequent steps are random in {−1, 1} if Xi 6= 0 and reflect to the previous step if Xi = 0 (i.e. Xi+1 = Xi−1). Now the sequence is not a martingale according to (2), since given Xi, Xi−1 . . . X0, if Xi = 0 then we know that Xi+1 = Xi−1. But requirement (1) holds, since given Xi = 0 we have no idea whether we arrived from the positives or negatives. Under both definitions we have Azuma’s inequality: ∗These notes summarize the material covered in class, usually skipping proofs, details, examples and so forth, and possibly adding some remarks, or pointers. The exercises are for self-practice and need not be handed in. In the interest of brevity, most references and credits were omitted.