3-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 $O(\sqrt{n})$. 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 $\Omega(\sqrt{n} \lg n)$ 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 $X_i$ are $k$-wise independent random variables with bounded $p$th moments. For general $k, p$, we determine the (asymptotically) maximum possible $p$th moment of the supremum of $X_1 + \dots + X_i$ over $1 \le i \le n$. We highlight the case $k = 4, p = 2$: here, we prove that the second moment of the furthest distance traveled is $O(\sum X_i^2)$. For this case, we only need the $X_i$'s to have bounded second moments and do not even need the $X_i$'s to be identically distributed. 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 B{\l}asiok.