arXiv subjects

José Ricardo G. Mendonça

Publications and source records attributed to José Ricardo G. Mendonça.

6 recordsLinked to original sources

arXiv2026

Near-unit-root persistence of symmetric and asymmetric stable autoregressive sequences

For an AR($1$) sequence with coefficient $0 < a < 1$, the probability of staying above zero decays exponentially in the number of steps; at the unit root $a=1$ it decays polynomially, with order $n^{-1/2}$ for symmetric increments. We study this transition for AR($1$) sequences driven by symmetric and asymmetric $α$-stable innovations. We write $Λ(a,α)$ for the symmetric exponential persistence rate and $Λ(a,α,ρ)$ for the rate of a strictly stable law with positivity parameter $ρ$. The entire chain admits an exact representation through a single stable Lévy process observed on a geometrically expanding time grid. In the symmetric case, comparison with continuous half-line survival gives $Λ(a,α) \leq \fracα{2}\log{(1/a)}$. For $0 < α< 2$, this bound disproves the stable specialization of a conjecture of Hinrichs, Kolb and Wachtel concerning regularly varying innovation tails. Combining stable closure under subsampling with a monotonicity coupling yields a lower bound of the same near-unit order. This proves $Λ(a,α) \asymp \log{(1/a)}$ as $a \uparrow 1$ and shows that $Λ(a,α)/\log{(1/a)}$ converges to a limit in $(0,α/2]$ equal to the supremum of that ratio over $0 < a < 1$. A Lamperti transformation reduces identification of this constant to a dense-sampling problem for a stationary stable Ornstein-Uhlenbeck process. Existing Gaussian theory determines the sharp value at $α=2$; for $0 < α< 2$, rescued crossings between observations remain the obstacle. For asymmetric strictly stable innovations with $α\neq 1$ and admissible $ρ\in(0,1)$, the corresponding bounds are $Λ(a,α,ρ) \leq α(1-ρ)\log{(1/a)}$ and $Λ(a,α,ρ) \asymp \log{(1/a)}$. At $α=2$, only the symmetric case $ρ=1/2$ occurs.

math.PR↗

arXiv2026

Skewness tunes the small-drift record rate of random walks and Lévy flights

A random walk with small positive drift $μ$ sets new records at a rate $λ(μ)$ that vanishes as $μ\to 0$. For Gaussian and strictly stable centered steps whose stable law $Y$ has index $1 < α\leq 2$ and positivity parameter $ρ= \mathbb{P}(Y>0)$, we find $λ(μ) \sim Kμ^{(1-ρ)/ν}$ as $μ\to 0$, where $ν=1-1/α$ and $K$ is explicit. Throughout their domains of attraction, the exponent persists, with a slowly varying factor replacing the constant $K$. The exponent is set by the asymmetry only through $ρ$, sweeping the interval $[1,\,1/(α-1)]$ as the skewness varies. For centered strictly stable steps, $ρ$ also governs the driftless record growth, $\langle R_{N}\rangle \sim N^ρ/Γ(1+ρ)$, which the small-drift law meets at the crossover where the drift takes over. The formula recovers the Gaussian linear law $λ(μ) \sim \sqrt{2}μ/σ$ and, for symmetric heavy tails, the power $μ^{α/2(α-1)}$. It follows from one Mellin transform of the harmonic sum in the Spitzer--Baxter identity, which factorizes into a kernel transform carrying the step distribution and a Riemann zeta function carrying the harmonic weights. Its poles deliver the leading law, its prefactor, and a correction ladder reproducing the known Gaussian and stable series, unifying diffusive, heavy-tailed, and skewed walks. The power law ends at the Cauchy point $α=1$, where $μ$ is a pure location shift: for the strictly Cauchy family the limiting rate vanishes and records accumulate sublinearly with a shift-dependent exponent.

cond-mat.stat-mech↗

arXiv2026

Finite-depth scaling and an exact Bernoulli-leaf identity for the min-plus process on the binary tree

The min-plus process is a stochastic coagulation-annihilation-type process on the binary tree, of interest in mathematics, physics, and computer science as a tractable instance of max-type recursive distributional equations. We carry out large Monte Carlo simulations at effective tree depths up to $N=60$ that provide finite-depth corroboration of the Beta(2,1) stretched-exponential limit for its root value $X_{N}$ at $p=1/2$, on the asymmetric $\sqrt{N}$ side of the random-homogeneous-systems classification recently introduced by Chen, Duquesne, and Shi and by Morfe. Off criticality, our simulations confirm the sub-critical closed form $\mathbb{P}(X_{\infty}=1)=(1-2p)/(1-p)$ within Monte Carlo error and document a super-critical mean growth exceeding the elementary $(2p)^{N}$ lower bound at the depths we reach. For a Bernoulli($q$)-initial-condition variant, we identify an elementary closed-form identity at $p=1/2$ that pins down the order parameter $\mathbb{P}(X_{N}=0)=q$ exactly, locates the absorbing-state phase transition at $p_{c}=1/2$ in the operator-mixing probability rather than in the initial-zero density, and shows that the conditional law on positives deforms substantially with $q$. Our simulations use a level-wise recursion and an FFT-based precomputed leaf table which reduce the effective simulation depth while preserving the recursive tree law and may be useful for the simulation of related recursive equations on large trees.

cond-mat.stat-mech↗

arXiv2026

On the normality of the concatenated Fibonacci constant

We study the concatenated Fibonacci constant $\mathcal{F} := 0.F_{1}F_{2}F_{3}\cdots = 0.11235813\cdots$, obtained by concatenating the Fibonacci numbers in the fractional part, and ask whether it is normal. We show that several classical sufficient conditions for normality by concatenation do not apply to the Fibonacci sequence because of its exponential growth, while a criterion of Pollack and Vandehey implies that the normality of $\mathcal{F}$ in base $10$ would follow if almost all Fibonacci numbers were $(\varepsilon,k)$-normal in base $10$. The Benford bias of leading digits and the Pisano periodicity of trailing digits are shown to contribute asymptotically negligible fractions of the total digits, isolating the distribution of the deep digits of large Fibonacci numbers as the remaining obstruction. Large-scale numerical experiments on the first $500{,}000$ Fibonacci numbers in bases $10$ and $2$ indicate that global single-digit counts and $k$-block statistics for $k = 2, 3, 4$ are compatible with i.i.d.-like fluctuations at the scales tested, and that a positional decomposition concentrates the visible structured deviation at the boundaries between consecutive Fibonacci numbers, while pooled interior blocks remain close to uniform. Our computations suggest that any obstruction to normality lies in the asymptotic behavior of the deep digits of $F_{n}$.

math.NT↗

arXiv2026

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

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.

cond-mat.stat-mech↗

arXiv2026

On the Golomb-Dickman constant under Ewens sampling

We define a generalized Golomb--Dickman constant $λ_θ$ as the limiting expected proportion of the longest cycle in random permutations under the Ewens measure with parameter $θ> 0$. Exploiting the independence properties of Kingman's Poisson process construction of the Poisson--Dirichlet distribution, we obtain an explicit integral representation for $λ_θ$ in terms of the exponential integral. The dependence of $λ_θ$ on $θ$ reflects the transition between regimes dominated by long cycles (small $θ$) and those with many small cycles (large $θ$). We also derive the asymptotic behavior of $λ_θ$ for small and large $θ$ and illustrate our results with numerical computations, Monte Carlo simulations of the Hoppe urn, and an application.

math.PR↗