Three‐wise independent random walks can be slightly unbounded

Abstract

Recently, many streaming algorithms have utilized generalizations of the fact that the expected maximum distance of any 4‐wise independent random walk on a line over n steps is O(n)$$ O\left(\sqrt{n}\right) $$ . In this paper, we show that 4‐wise independence is required for all of these algorithms, by constructing a 3‐wise independent random walk with expected maximum distance Ω(nlgn)$$ \Omega \left(\sqrt{n}\lg n\right) $$ from the origin. We prove that this bound is tight for the first and second moment, and also extract a surprising matrix inequality from these results. Next, we consider a generalization where the steps Xi$$ {X}_i $$ are k‐wise independent random variables with bounded pth moments. We highlight the case k=4,p=2$$ k=4,p=2 $$ : here, we prove that the second moment of the furthest distance traveled is O∑Xi2$$ O\left(\sum {X}_i^2\right) $$ . This implies an asymptotically stronger statement than Kolmogorov's maximal inequality that requires only 4‐wise independent random variables, and generalizes a recent result of Błasiok.