Pointwise Majorization for sub-Weibull and Mixed Tail Processes with Applications in Quadratic Chaos and Ergodic Diffusions

Classical chaining controls an indexed stochastic process through a single worst-case bound, which can obscure substantial variation across the index set. We establish the first simultaneous pointwise majorization theory for Banach-valued processes with sub-Weibull or two-metric mixed-tail increments. For an anchored sub-Weibull process on a separable index space, write $v(t):=d(t,t_0)$. Given a reference measure $μ$, the envelope at $t$ is governed by the pointwise Fernique-Talagrand functional of order $α$, $Φ_{μ,d}^{(α)}(t):=\int_0^{4v(t)}(\log\frac{1}{μ(B_d(t,r))})^{1/α}dr$. $\forall δ\in(0,1)$, we obtain that $$ \mathbb{P}(\|Z_t\|\lesssim\{Φ_{μ,d}^{(α)}(t)+v(t)(\log(e/δ))^{1/α}\},\forall t)\ge 1-δ. $$ Our bound is determined by the pointwise complexity $Φ_{μ,d}^{(α)}$ rather than a global quantity. The result holds for every $α>0$ and does not involve dyadic logarithmic terms from peeling. For mixed tail processes, with fixed measures $μ_1,μ_2$ and $v_j(t):=d_j(t,t_0)$, $Φ_j(t):=\int_0^{4v_j(t)}(\log\frac{1}{μ_j(B_{d_j}(t,r))})^{1/α_j}dr, j=1,2$, for any $δ\in(0,1)$, we show that $$\mathbb{P}(\|Z_t\|\lesssim\sum_{j=1}^2\{Φ_j(t)+v_j(t)(\log\frac{e}δ)^{1/α_j}\},\forall t)\ge 1-δ.$$ Although the two regimes are coupled in the mixed tail condition, each retains its own pseudo-metric, reference measure, pointwise Fernique-Talagrand functional, and tail exponent. The proof tracks the index-wise costs of measure-generated admissible chains and synchronizes them through a nested common refinement. For applications, we derive matrix-specific bounds for centered quadratic chaos under pseudo-metrics induced by the operator and Frobenius norms, and observable-specific finite-time bounds for diffusion empirical processes.

The Exact Worst-Case Tail Probability under Bounded Kurtosis

We determine exactly what a kurtosis bound buys for one-sided tail control. For the class $\mathcal{C}(κ)$ of real random variables with mean $0$, variance $1$, and fourth moment at most $κ$, the skewness left free, we compute the worst-case tail probability $V_1(t,κ)=\sup_{X\in\mathcal{C}(κ)}\mathbb{P}(X\geq t)$ for every threshold $t>0$ and every $κ\geq 1$. The answer is a four-regime map: a Cantelli tongue $b(κ)\le t\le c(κ)$ on which the two-moment bound $1/(1+t^2)$ remains tight and the kurtosis constraint is worthless; a tail regime $t\geq c(κ)$ with the closed form $V_1=(κ-1)/((t^2-1)^2+κ-1)$; a plateau regime, present only for $κ\le 3/2$, on which the worst case freezes and the value does not depend on $t$; and a central regime described exactly by an explicit algebraic system, provably admitting no closed form in nested square roots. Beyond $c(κ)$ the one-sided and two-sided worst cases coincide: Cantelli's improvement over Chebyshev is annihilated by fourth-moment information. The minimal degree of a sum-of-squares proof of the tight bound is $2$ on the closed tongue and $4$ everywhere else, an exact phase diagram of proof degree. Every closed-form regime carries an explicit dual certificate and an explicit extremal distribution, re-verified on parameter grids by an independent checker in exact arithmetic. The closed forms invert to exact worst-case quantiles, sharpen a median-of-means constant, and give the exact per-direction tail available to degree-4 reasoning under certifiable kurtosis. We found the map through an AI-guided search around the certifying pipeline, LemmaForge, which is validated on classical benchmarks, independently reproduces the symmetric-slice bound of Zelen (1954), and recovers the $2\sqrt{3}-3$ constant of He, Zhang, and Zhang (2010) at $t=0$.

History estimation in random recursive trees: Pointwise approach via iterated Jordan centralities

We study the problem of estimating the arrival times of vertices in a uniform random recursive tree from its unlabeled structure. We adopt a pointwise perspective and analyze the distribution of the relative estimation error, and derive tail bounds that are uniform in both the vertex and the tree size. For the ranking induced by Jordan centrality, the probability that the estimate exceeds the true arrival time by a factor $S$ decays on the order of $1/S$, while the probability of underestimating the arrival time by a factor $1/S$ decays exponentially in $S$. We introduce a refined centrality measure whose overestimation tail decays on the order of $(\log S)/S^{2}$, at the cost of a heavier lower tail of order $1/S^{2}$. These results reveal a tradeoff between upper- and lower-tail performance in arrival-time estimation that is invisible to the previously studied risk functional. Nevertheless, the refined centrality measure attains the optimal order of the risk for all its parameter values.

Non-asymptotic Tail Bounds for the Kostlan--Shub--Smale Field: Tensor PCA and Spherical $k$-Spin Complexity

This paper builds a hierarchy of explicit, non-asymptotic tail bounds for the supremum of the Kostlan--Shub--Smale (KSS) random field on the sphere, and applies it to two problems: Spiked Tensor PCA and the landscape of the spherical $k$-spin model. For Tensor PCA, we study the non-asymptotic statistical limits of estimating a rank-$R$ symmetric signal tensor of order~$k\ge 3$ and dimension~$d\ge 3$ from a single Gaussian observation at signal-to-noise ratio~$λ$, through the \emph{profile maximum likelihood estimator}, the MLE restricted to normalized rank-$R$ tensors of coherence at least~$κ$. Our analysis uses a single reduction: a deterministic geometric inequality (the Tube Method) and a rank-reduction step bound the estimation error by the supremum of the canonical KSS field, which the Kac--Rice formula turns into a Gaussian integral against the expected absolute characteristic polynomial of a shifted Gaussian Orthogonal Ensemble, controlled in turn by the four explicit tail bounds of our hierarchy (three from a Mehta--Fyodorov representation, one from a Ben Arous--Dembo--Guionnet large deviation). The same reduction yields two results, each with explicit constants. For estimation, a finite-$(k,d)$ error bound recovers the asymptotically optimal rate~$\sqrt{d\log k}$ of Perry, Wein and Bandeira, with explicit dependence on the rank~$R$ and the coherence~$κ$. For the landscape, a two-sided non-asymptotic bracketing of the annealed complexity of the spherical $k$-spin Hamiltonian recovers the Auffinger--Ben Arous--Černý complexity function in the high-dimensional limit.