  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.
