arXiv subjects

J. E. Paguyo

Publications and source records attributed to J. E. Paguyo.

11 recordsLinked to original sources

arXiv2026

Asymptotic behavior of clusters in hierarchical species sampling models

Consider a random sample of size $N$ from a hierarchical species sampling model. In this paper, we study the large $N$ asymptotic behavior of the number ${\bf K}_N$ of clusters in the random sample and the number ${\bf \widetilde M}_{\ell,N}$ of clusters represented by exactly $\ell$ latent first-level clusters in the first level of the hierarchical model. In particular, we establish almost sure and $L^p$ convergence for ${\bf \widetilde M}_{\ell,N}$, Gaussian fluctuations and the law of the iterated logarithm for ${\bf K}_N$, and large deviation principles for both ${\bf K}_N$ and ${\bf \widetilde M}_{\ell,N}$. Our approach relies on a random sample size (or random index) representation of the number of clusters through the corresponding non-hierarchical species sampling model.

math.PR↗

arXiv2026

Analysis of a twisted Bose-Einstein Markov chain with applications to sampling Catalan structures

We analyze a twisted Bose-Einstein Markov chain on $[k]^n$ arising from the Diaconis-Zhong twisted Burnside process. We derive explicit formulas for the transition kernel and stationary distribution of both the original chain and its lumped process, and we identify this chain as a special case of Crane's cut-and-paste process. We establish bounds on the mixing time and determine the second-largest eigenvalue of the original chain and its lumped process. Complementing the fixed-$k$ cutoff theorem of Crane and Lalley, our bounds show that both chains mix in $Θ(\log n)$ steps when $k$ grows as a fixed positive power of $n$. For the non-twisted Bose-Einstein Markov chain, this resolves a conjecture of Diaconis. As an application, we study the Burnside processes on parking functions and labeled Dyck paths, which give novel Markov chain Monte Carlo algorithms for sampling an increasing parking function and a Dyck path, respectively, approximately uniformly at random. We show that these chains are rapidly mixing, with mixing times of $Θ(\log n)$.

math.PR↗

arXiv2026

Central limit theorem for the homozygosity of the hierarchical Pitman-Yor process

The hierarchical Pitman-Yor process is a discrete random measure used as a prior in Bayesian nonparametrics. It is motivated by the study of groups of clustered data exhibiting power law behavior. Our focus in this paper is on the Gaussian behavior of a family of statistics, namely the power sum symmetric polynomials for the vector of weights of the process, as the concentration parameters tend to infinity. We establish a central limit theorem and obtain explicit representations for the asymptotic variance, with the latter clearly showing the impact of each component in the hierarchical structure. These results are crucial for understanding the asymptotic behavior of the sampling formulas associated with the process. In comparison with the known results for the hierarchical Dirichlet process, the results for the hierarchical Pitman-Yor process are mathematically more challenging and structurally more revealing of power law behavior.

math.PR↗

arXiv2026

Limit distributions for cycles of random parking functions

We study the asymptotic behavior of cycles of uniformly random parking functions. Our results are multifold: we obtain an explicit formula for the number of parking functions with a prescribed number of cyclic points and show that the scaled number of cyclic points of a random parking function is asymptotically Rayleigh distributed; we establish the classical trio of limit theorems (law of large numbers, central limit theorem, large deviation principle) for the number of cycles in a random parking function; we also compute the asymptotic mean of the length of the $r$th longest cycle in a random parking function for all valid $r$. A variety of tools from probability theory and combinatorics are used in our investigation. Corresponding results for the class of prime parking functions are obtained.

math.PR↗

arXiv2026

Mixing times of a Burnside process Markov chain on set partitions

Let $X$ be a finite set and let $G$ be a finite group acting on $X$. The group action splits $X$ into disjoint orbits. The Burnside process is a Markov chain on $X$ which has a uniform stationary distribution when the chain is lumped to orbits. We consider the case where $X = [k]^n$ with $k \geq n$ and $G = S_k$ is the symmetric group on $[k]$, such that $G$ acts on $X$ by permuting the value of each coordinate. The resulting Burnside process gives a novel algorithm for sampling a set partition of $[n]$ uniformly at random. We obtain bounds on the mixing time and show that the chain is rapidly mixing. For the case $k < n$, the algorithm corresponds to sampling a set partition of $[n]$ with at most $k$ blocks, and we obtain a mixing time bound which is independent of $n$. Along the way, we obtain explicit formulas for the transition probabilities and bounds on the second largest eigenvalue for both the original process and the lumped chain.

math.PR↗

arXiv2025

Central limit theorems associated with the hierarchical Dirichlet process

The hierarchical Dirichlet process is a discrete random measure used as a prior in Bayesian nonparametrics and motivated by the study of groups of clustered data. We study the asymptotic behavior of the power sum symmetric polynomials for the vector of weights of the hierarchical Dirichlet process as the concentration parameters tend to infinity. We establish central limit theorems and obtain explicit representations for the asymptotic variances, with the latter clearly showing the impact of the hierarchical structure. These objects are related to the homozygosity in population genetics, the Simpson diversity index in ecology, and the Herfindahl-Hirschman index in economics.

math.PR↗

arXiv2024

Central limit theorem for crossings in randomly embedded graphs

We consider the number of crossings in a random embedding of a graph, $G$, with vertices in convex position. We give explicit formulas for the mean and variance of the number of crossings as a function of various subgraph counts of $G$. Using Stein's method and size-bias coupling, we establish an upper bound on the Kolmogorov distance between the distribution of the number of crossings and a standard normal random variable. We also consider the case where $G$ is a random graph and obtain a Kolmogorov bound between the distribution of crossings and a Gaussian mixture distribution. As applications, we obtain central limit theorems with convergence rates for the number of crossings in random embeddings of matchings, path graphs, cycle graphs, disjoint union of triangles, random $d$-regular graphs, and mixtures of random graphs.

math.PR↗

arXiv2023

Fixed points, descents, and inversions in parabolic double cosets of the symmetric group

We consider statistics on permutations chosen uniformly at random from fixed parabolic double cosets of the symmetric group. We show that the distribution of fixed points is asymptotically Poisson and establish central limit theorems for the distribution of descents and inversions. Our proofs use Stein's method with size-bias coupling and dependency graphs, which also gives convergence rates for our distributional approximations. As applications of our size-bias coupling and dependency graph constructions, we obtain concentration of measure results on the number of fixed points, descents, and inversions.

math.PR↗

arXiv2023

Convergence rates of limit theorems in random chord diagrams

We study the asymptotic distributions of the number of crossings and the number of simple chords in a random chord diagram. Using size-bias coupling and Stein's method, we obtain bounds on the Kolmogorov distance between the distribution of the number of crossings and a standard normal random variable, and on the total variation distance between the distribution of the number of simple chords and a Poisson random variable. As an application, we provide explicit error bounds on the number of chord diagrams containing no simple chords.

math.PR↗

arXiv2022

Cycle structure of random parking functions

We initiate the study of the cycle structure of uniformly random parking functions. Using the combinatorics of parking completions, we compute the asymptotic expected value of the number of cycles of any fixed length. We obtain an upper bound on the total variation distance between the joint distribution of cycle counts and independent Poisson random variables using a multivariate version of Stein's method via exchangeable pairs. Under a mild condition, the process of cycle counts converges in distribution to a process of independent Poisson random variables.

math.PR↗

arXiv2017

Maximal spanning time for neighborhood growth on the Hamming plane

We consider a long-range growth dynamics on the two-dimensional integer lattice, initialized by a finite set of occupied points. Subsequently, a site $x$ becomes occupied if the pair consisting of the counts of occupied sites along the entire horizontal and vertical lines through $x$ lies outside a fixed Young diagram $\mathcal{Z}$. We study the extremal quantity $μ(\mathcal{Z})$, the maximal finite time at which the lattice is fully occupied. We give an upper bound on $μ(\mathcal{Z})$ that is linear in the area of the bounding rectangle of $\mathcal{Z}$, and a lower bound $\sqrt{s-1}$, where $s$ is the side length of the largest square contained in $\mathcal{Z}$. We give more precise results for a restricted family of initial sets, and for a simplified version of the dynamics.

math.CO↗