  We numerically estimate the leading asymptotic behavior of the length $L_{n}$
of the longest increasing subsequence of random walks with step increments
following Student's $t$-distribution with parameter in the range $1/2 \leq \nu
\leq 5$. We find that the expected value $\mathbb{E}(L_{n}) \sim
n^{\theta}\ln{n}$ with $\theta$ decreasing from $\theta(\nu=1/2) \approx 0.70$
to $\theta(\nu \geq 5/2) \approx 0.50$. For random walks with distribution of
step increments of finite variance ($\nu > 2$), this confirms previous
observation of $\mathbb{E}(L_{n}) \sim \sqrt{n}\ln{n}$ to leading order. We
note that this asymptotic behavior (including the subleading term) resembles
that of the largest part of random integer partitions under the uniform measure
and that, curiously, both random variables seem to follow Gumbel statistics. We
also provide more refined estimates for the asymptotic behavior of
$\mathbb{E}(L_{n})$ for random walks with step increments of finite variance.
