3-wise Independent Random Walks can be Slightly Unbounded

TL;DR

This paper demonstrates that 3-wise independence is insufficient for certain random walk bounds, establishing the necessity of 4-wise independence for streaming algorithms, and extends maximal inequalities to broader independence settings.

Contribution

It proves the tightness of 4-wise independence for maximum distance bounds and generalizes maximal inequalities to k-wise independent variables with bounded moments.

Findings

3-wise independence leads to larger maximum distances than 4-wise independence.

The second moment of the maximum distance is bounded by the sum of squared steps for 4-wise independent variables.

The results imply stronger maximal inequalities under weaker independence assumptions.

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 $O(\sqrt{n})$. 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.