We investigate the behavior of the length of the longest weakly increasing subsequences (weak LIS) of $n$-step random walks with nonzero integer increments $k = \pm 1, \pm 2, \dots$ given by a symmetric heavy tailed mass distribution proportional to $|k|^{-1-α}$ for several values of the real parameter $α> 0$ together with that of the simple random walk ($k=\pm 1$), to which the $n$-step heavy tailed walks reduce when $α$ grows large enough that step jumps beyond $\pm 1$ become essentially absent on the scale of $n$. By means of exploratory fits, weighted nonlinear least squares, and nested-model comparisons, we found that the sample average length $\langle{L_{n}}\rangle$ scales like $\langle{L_{n}}\rangle \sim \sqrt{n}\log{n}$ when the distribution of increments has finite variance ($α> 2$) and $\langle{L_{n}}\rangle \sim n^θ$ with a varying exponent $θ> 0.5$ when the variance is infinite ($α\leq 2$). Distributional diagnostics indicate that the bulk of the $L_{n}$ distribution is very well-approximated by a lognormal model, though systematic deviations are observed in the tails. Our results corroborate and expand upon previous results for the LIS of other types of heavy-tailed random walks and raise a conjecture as to whether the distribution of $L_{n}$ is given, or can be effectively described, by a lognormal distribution.

翻译：我们研究了$n$步非零整数增量$k = \pm 1, \pm 2, \dots$的随机游走的最长弱递增子序列（weak LIS）长度的行为。这些增量由对称重尾质量分布给出，该分布与$|k|^{-1-α}$成正比，涉及实参数$α>0$的多个取值；同时研究了简单随机游走（$k=\pm 1$）的情形——当$α$足够大以致在$n$的尺度上$\pm 1$以外的步长跳跃基本消失时，$n$步重尾游走退化为该情形。通过探索性拟合、加权非线性最小二乘法和嵌套模型比较，我们发现：当增量分布具有有限方差（$α>2$）时，样本平均长度$\langle{L_{n}}\rangle$按$\langle{L_{n}}\rangle \sim \sqrt{n}\log{n}$标度；当方差无限（$α\leq2$）时，$\langle{L_{n}}\rangle \sim n^θ$，其中指数$θ>0.5$可变。分布诊断表明，$L_{n}$分布的主体部分非常接近对数正态模型，尽管在尾部观测到系统性偏差。我们的结果佐证并扩展了先前关于其他类型重尾随机游走LIS的结果，并提出一个猜想：$L_{n}$的分布是否由对数正态分布给出，或能否用其有效描述。