arXiv subjects

Víctor J. Maciá

Publications and source records attributed to Víctor J. Maciá.

11 recordsLinked to original sources

arXiv2026

Wiman-Valiron inequalities in the unit disk outside sets of finite logarithmic measure

We give affirmative answers to both parts of Question 2.6 posed by Grosse-Erdmann (2025) concerning Wiman-Valiron inequalities in the unit disk. For every unbounded analytic function in the disk, we establish the proposed iterated-logarithm inequalities outside exceptional sets of finite logarithmic measure. The corresponding power estimate is a corollary. Both conclusions follow from a variance bound for Khinchin families and a classical estimate for their largest atom. The multiplicative constants in the main inequalities can be chosen absolute. We also obtain a disk analogue of Rosenbloom's composition estimate, with an explicit boundary prefactor. The key step combines a boundary change of variable with a monotone auxiliary function whose derivative is exactly the variance of a rescaled member of the Khinchin family. A classical example shows that the leading logarithmic exponent $1/2$ cannot be decreased.

math.CV↗

arXiv2026

Uniform integrability of the distance to the nearest leaf in random trees

We study the distance from the root to the nearest leaf, the analogous quantity for a uniformly chosen vertex, and its protection number, in size-conditioned simply generated trees. We prove a uniform exponential tail bound for each of these quantities, valid for arbitrary offspring distributions. As a consequence, these random variables are uniformly integrable of every order. This yields convergence of all moments to those of the corresponding local limit. The argument is probabilistic and unified across the three quantities.

math.PR↗

arXiv2026

Limit laws of random simplex tree-child networks

We prove that the longer and shorter Sackin indices of a uniformly random simplex tree-child network with $n$ taxa admit joint distributional limits after rescaling by $n^{-7/4}$. The limiting distributions are described by functionals of a Brownian excursion. We also identify the limiting law of the height after rescaling by $n^{-3/4}$, thereby answering a question of Zhang~(2022). Moreover, we establish sharp tail bounds for the height, which imply convergence of all moments in the above distributional limits. We further obtain a scaling limit for the entire height profile of the leaves. Finally, we determine the local limits of large simplex networks around the fixed root, a uniformly random vertex, and a uniformly random leaf.

math.CO↗

arXiv2026

Power Partitions and Hayman Functions

We prove, within the probabilistic framework of Khinchin families, that the generating function $P_k$ of partitions into $k$-th powers is strongly Gaussian in the sense of Báez-Duarte, and even further that it is a Hayman function. Thus the Hardy--Ramanujan asymptotic formula for the number $p_k(n)$ of partitions of $n$ into $k$-th powers which reads \[ p_k(n) \sim \frac{α_k}{n^{(3k+1)/(2k+2)}} \exp\!\Big(β_k\, n^{1/(k+1)}\Big), \qquad n\to\infty, \] where $α_k$ and~$β_k$ are explicit constants depending only on $k$, follows directly from Hayman's asymptotic formula for strongly Gaussian power series. The proof of strong Gaussianity of $P_k$ combines a Gaussianity criterion for Khinchin families with certain bounds of Tenenbaum, Wu and Li on the generating function; the asymptotic formula is recovered by computing asymptotic approximations of the mean and variance of the associated family. Analogous results are presented for the generating function $Q_k$ of partitions into distinct $k$-th powers.

math.PR↗

arXiv2026

Galton-Watson processes, simple varieties of trees and Khinchin families

In this note, we introduce a unified analytic framework that connects simple varieties of trees, Bienayme-Galton-Watson processes and Khinchin families. Using Lagrange's inversion formula, we derive new coefficient-based expressions for extinction probabilities and reinterpret them as boundary phenomena tied to the domain of the inverse of the solution to Lagrange's equation. This perspective reveals an additional link between combinatorial and probabilistic models, simplifying classical arguments and yielding new results. It also leads to a computationally efficient method for simulating Galton-Watson processes via power series coefficients.

math.CV↗

arXiv2025

The Theory of Khinchin Families

The theory of Khinchin families connects Probability Theory and Complex Analysis. Along this PhD thesis, we exploit this connection to obtain, using a variety of local central limit theorems, asymptotic formulas for the coefficients of analytic functions with non-negative coefficients. We give criteria for a power series to have certain specific, and in most cases, computable, asymptotic formula. The theory of Khinchin families was initiated and is strongly influenced by the work of Walter K. Hayman, Paul C. Rosembloom, and Luis Báez-Duarte. We can assign to each power series $f$ in the class $\mathcal{K}$, having radius of convergence $R>0$, a family of random variables $(X_t)_{[0,R)}$. This theory examines the behavior of this family of random variables, and also of its normalized version, as $t \uparrow R$. This thesis aims to harmonize the various asymptotic formulas for combinatorial or probabilistic objects within a single framework. With this aim we have extended some of the classes of functions and also studied the coefficients of large powers of analytic functions. One of our main goals was to consolidate and develop some aspects of a comprehensive theory: providing a guidebook for combinatorialists or probabilists, a series of results, or basic criteria, where they can follow a step-by-step process, verify if certain straightforward conditions are met, and then derive an asymptotic formula for the object of study. However, this is not the sole primary goal; we are also profoundly interested in the functional theoretical properties of these classes of functions and their connections to the respective Khinchin families (such as the case of $f \in \mathcal{K}$ being an entire function).

math.CV↗

arXiv2025

The distance to the border of a random tree

Given a Galton-Watson process conditioned to have total progeny equal to $n$, we study the asymptotic probability that this conditioned Galton-Watson process has distance to the border bigger or equal than $k$, as the number of nodes $n \rightarrow \infty$. A problem which is akin to this one was solved by Rényi and Szekeres for Cayley trees, de Bruijn, Knuth, and Rice for plane trees and Flajolet, Gao, Odlyzko, and Richmond for binary trees. The distance to the border is dual, in a certain sense, to the height. The first of these distances is the minimum of the distances from the root to the leaves. The second is the maximum of the distances from the root to the leaves. These are two extreme complementary cases.

math.PR↗

arXiv2025

Some Gaussianity criteria for Khinchin families

In this note we establish simple and verifiable analytical conditions for a power series f in the class K, and its associated Khinchin family, to be Gaussian. We give several moment criteria for general power series with non-negative coefficients in K and also provide several applications of these criteria.

math.PR↗

arXiv2024

Growth of power series with nonnegative coefficients, and moments of power series distributions

Any power series with nonnegative coefficients has an associated family of probability distributions supported on the nonnegative integers. There is a close connection between the function theoretic properties of the power series and the moments of the family of distributions. In this paper, we describe that interplay, provide simpler proofs of some known results by emphasizing the probabilistic perspective, and present some new theorems.

math.PR↗

arXiv2023

Large Powers asymptotics, Khinchin families and Lagrangian distributions

This paper delves on the versatility of the theory of Khinchin families for asymptotic estimation. We show that in combination with Local Central Limit theorems for lattice variables, Khinchin families furnish a convenient and unified framework to deal with asymptotic results of the coefficients of large powers of power series. We revisit in the present paper this classical theme from that point of view, obtaining clean new proofs and a number of new results. Asymptotic results for the coefficients of solutions of Lagrange's equation fall naturally into this combined framework. We provide a direct proof of an extension of the Otter and Meir-Moon asymptotic formula as well as asymptotic results for families of Lagrangian probability distributions.

math.PR↗

arXiv2022

Khinchin families, set constructions, partitions and exponentials

In this paper, we give a simple criterion to verify that functions of the form $e^g$ are in the Hayman class when $g$ is a power series with nonnegative coefficients. Thus, using the Hayman and Báez-Duarte formulas, we obtain asymptotics for the coefficients of generating functions that arise in many examples of set construction in analytic combinatorics. This new criterion greatly simplifies that obtained previously by the authors.

math.CO↗