Three‐wise independent random walks can be slightly unbounded

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.