Longest weakly increasing subsequences of discrete random walks on the integers with heavy tailed distribution of increments

Abstract

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 zero-mean, 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 random walks tends when $α> (1+o(1))\log_{2}{n}$. By means of exploratory fits, weighted nonlinear least squares, and ANOVA model comparison, 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.

Figures (13)

• Figure 1: Sample random walks (\ref{['eq:sym']}) and their associated weak LIS (circles) for different values of $\alpha$ (see Eq. (\ref{['eq:prb']})) and a simple random walk (SRW). Note the different vertical scales. For $n=300$, the random walk with $\alpha=10$ and the simple random walk are virtually indistinguishable, since in this case the probability that some step increment or decrement of the walk is larger than $1$ becomes smaller than $10^{-3}$; in panel (c), for instance, there is none.
• Figure 2: Leading exponent $\theta$ (see (\ref{['eq:heavy']})) in the asymptotic behavior of the LIS of random walks with symmetric continuous heavy tailed distribution of increments $\phi_{\alpha}(\left|x\right| \gg 1) \sim \left|x\right|^{-1-\alpha}$ in the range $0 < \alpha \leq 3$. The point at $\alpha=0$ represents the value of $\theta$ found for the LIS of the ultra-fat random walk. When $\alpha \geq 2$ the values of $\theta$ approach $0.5$ and the asymptotic behavior of the LIS acquires a logarithmic factor, see (\ref{['eq:short']}).
• Figure 3: Effective exponent $\theta_{\text{eff}}(n) = \Delta\log\langle L_{n}\rangle /\Delta\log n$ of the weak LIS of discrete heavy tailed random walks (\ref{['eq:sym']}) as a function of $\log n$, evaluated at the midpoints of consecutive pairs of walk lengths. For $\alpha \leq 1$, $\theta_{\text{eff}}$ is approximately constant, consistent with a pure power law $\langle L_{n}\rangle \sim n^{\theta}$. For $\alpha \geq 2$ and the simple random walk (SRW), $\theta_{\text{eff}}$ decreases systematically, indicating the presence of a subleading logarithmic correction. The dashed line marks $\theta = 1/2$.
• Figure 4: Ratio plots for $\alpha \geq 2$ and the simple random walk (SRW). (a) Ratio $\langle{L_{n}}\rangle/(\sqrt{n}\mkern1mu\log{n})$ as a function of $\log{n}$. The monotonic increase indicates that the asymptotic regime $\langle{L_{n}}\rangle \sim b\sqrt{n}\mkern1mu\log{n}$ has not yet been reached; the intercept $a$ in Model II (Table \ref{['tab:adjIIb']}) is still contributing at these values of $n$. (b) Ratio $\langle{L_{n}}\rangle/n^{\overline{\theta}}$ as a function of $\log{n}$, where $\overline{\theta}$ is the Model I exponent from Table \ref{['tab:adjI']} for each $\alpha$. The approximately linear growth with $\log{n}$ reveals a residual logarithmic correction not captured by the power law alone.
• Figure 5: Stability analysis for $\alpha \geq 2$ and the SRW. The estimated exponent $\hat{\theta}$ of a pure power law fit to the data with $n \geq n_{\min}$ is plotted as a function of $\log{n_{\min}}$. The systematic downward drift toward $\theta = 1/2$ (dashed line) indicates that the elevated $\hat{\theta}$ obtained from fitting all data points is an artifact of the logarithmic correction. Error bars indicate $\pm 2$ standard errors.