arXiv subjects

Takashi Owada

Publications and source records attributed to Takashi Owada.

At least 19 recordsLinked to original sources

arXiv2026

Layered Hill estimator for extreme data in clusters

A new estimator is proposed for estimating the tail exponent of a heavy-tailed distribution. This estimator, referred to as the layered Hill estimator, is a generalization of the traditional Hill estimator, building upon a layered structure formed by clusters of extreme values. We argue that the layered Hill estimator provides a robust alternative to the traditional approach, exhibiting desirable asymptotic properties such as consistency and asymptotic normality for the tail exponent. Both theoretical analysis and simulation studies demonstrate that the layered Hill estimator shows significantly better and more robust performance, particularly when a portion of the extreme data is missing.

math.ST↗

arXiv2026

Stable and Fréchet limit theorem for subgraph functionals in the hyperbolic random geometric graph

We study the fluctuations of subgraph counts in hyperbolic random geometric graphs on the $d$-dimensional Poincaré ball in the heterogeneous, heavy-tailed degree regime. In a hyperbolic random geometric graph whose vertices are given by a Poisson point process on a growing hyperbolic ball, we consider two basic families of subgraphs: star shape counts and clique counts, and we analyze their global counts and maxima over the vertex set. Working in the parameter regime where a small number of vertices close to the center of the Poincaré ball carry very large degrees and act as hubs, we establish joint functional limit theorems for suitably normalized star shape and clique count processes together with the associated maxima processes. The limits are given by a two-dimensional dependent process whose components are a stable Lévy process and an extremal Fréchet process, reflecting the fact that a small number of hubs dominates both the total number of local subgraphs and their extremes. As an application, we derive fluctuation results for the global clustering coefficient, showing that its asymptotic behavior is described by the ratio of the components of a bivariate Lévy process with perfectly dependent stable jumps.

math.PR↗

arXiv2025

Normal approximation for subgraph counts in age-dependent random connection models

We study normal approximation of subgraph counts in a model of spatial scale-free random networks known as the age-dependent random connection model. In the light-tailed regime where only moments of order $(2 + \varepsilon)$ are finite, we study the asymptotic normality of both clique and subtree counts. For clique counts, we establish a multivariate quantitative normal approximation result through the Malliavin-Stein method. In the more delicate case of subtree counts, we obtain distributional convergence based on a central limit theorem for sequences of associated random variables.

math.PR↗

arXiv2024

Degree counts in random simplicial complexes of the preferential attachment type

We extend the classical preferential attachment random graph model to random simplicial complexes. At each stage of the model, we choose one of the existing $k$-simplices with probability proportional to its $k$-degree. The chosen $k$-simplex then forms a $(k+1)$-simplex with a newly arriving vertex. We establish a strong law of large numbers for the degree counts across multiple dimensions. The limiting probability mass function is expressed as a mixture of mass functions of different types of negative binomial random variables. This limiting distribution has power-law characteristics and we explore the limiting extremal dependence of the degree counts across different dimensions in the framework of multivariate regular variation. Finally, we prove multivariate weak convergence, under appropriate normalization, of degree counts in different dimensions, of ordered $k$-simplices. The resulting weak limit can be represented as a function of independent linear birth processes with immigration.

math.PR↗

arXiv2024

Limit theorems under heavy-tailed scenario in the age dependent random connection models

This paper considers limit theorems associated with subgraph counts in the age-dependent random connection model. First, we identify regimes where the count of sub-trees converges weakly to a stable random variable under suitable assumptions on the shape of trees. The proof relies on an intermediate result on weak convergence of associated point processes towards a Poisson point process. Additionally, we prove the same type of results for the clique counts. Here, a crucial ingredient includes the expectation asymptotics for clique counts, which itself is a result of independent interest.

math.PR↗

arXiv2023

Limit theorems for critical faces above the vanishing threshold

We investigate convergence of point processes associated with critical faces for a Čech filtration built over a homogeneous Poisson point process in the $d$-dimensional flat torus. The convergence of our point process is established in terms of the $\mathcal M_0$-topology, when the connecting radius of a Čech complex decays to $0$, so slowly that critical faces are even less likely to occur than those in the regime of threshold for homological connectivity. We also obtain a series of limit theorems for positive and negative critical faces, all of which are considerably analogous to those for critical faces.

math.PR↗

arXiv2023

Limit theorems for high-dimensional Betti numbers in the multiparameter random simplicial complexes

We consider the multiparameter random simplicial complex on a vertex set $\{ 1,\dots,n \}$, which is parameterized by multiple connectivity probabilities. Our key results concern the topology of this complex of dimensions higher than the critical dimension. We show that the higher-dimensional Betti numbers satisfy strong laws of large numbers and central limit theorems. Moreover, lower tail large deviations for these Betti numbers are also discussed. Some of our results indicate an occurrence of phase transitions in terms of the scaling constants of the central limit theorem, and the exponentially decaying rate of convergence of lower tail large deviation probabilities.

math.PR↗

arXiv2023

Large deviations for hyperbolic $k$-nearest neighbor balls

We prove a large deviation principle for the point process of large Poisson $k$-nearest neighbor balls in hyperbolic space. More precisely, we consider a stationary Poisson point process of unit intensity in a growing sampling window in hyperbolic space. We further take a growing sequence of thresholds such that there is a diverging expected number of Poisson points whose $k$-nearest neighbor ball has a volume exceeding this threshold. Then, the point process of exceedances satisfies a large deviation principle whose rate function is described in terms of a relative entropy. The proof relies on a fine coarse-graining technique such that inside the resulting blocks the exceedances are approximated by independent Poisson point processes.

math.PR↗

arXiv2022

Large deviations for the volume of $k$-nearest neighbor balls

This paper develops the large deviations theory for the point process associated with the Euclidean volume of $k$-nearest neighbor balls centered around the points of a homogeneous Poisson or a binomial point processes in the unit cube. Two different types of large deviation behaviors of such point processes are investigated. Our first result is the Donsker-Varadhan large deviation principle, under the assumption that the centering terms for the volume of $k$-nearest neighbor balls grow to infinity more slowly than those needed for Poisson convergence. Additionally, we also study large deviations based on the notion of $\mathcal M_0$-topology, which takes place when the centering terms tend to infinity sufficiently fast, compared to those for Poisson convergence. As applications of our main theorems, we discuss large deviations for the number of Poisson or binomial points of degree at most $k$ in a geometric graph in the dense regime.

math.PR↗

arXiv2022

Large deviation principle for geometric and topological functionals and associated point processes

We prove a large deviation principle for the point process associated to $k$-element connected components in $\mathbb R^d$ with respect to the connectivity radii $r_n\to\infty$. The random points are generated from a homogeneous Poisson point process, so that $(r_n)_{n\ge1}$ satisfies $n^kr_n^{d(k-1)}\to\infty$ and $nr_n^d\to0$ as $n\to\infty$ (i.e., sparse regime). The rate function for the obtained large deviation principle can be represented as relative entropy. As an application, we deduce large deviation principles for various functionals and point processes appearing in stochastic geometry and topology. As concrete examples of topological invariants, we consider persistent Betti numbers of geometric complexes and the number of Morse critical points of the min-type distance function.

math.PR↗

arXiv2022

Large deviations for subcomplex counts and Betti numbers in multi-parameter simplicial complexes

We consider the multi-parameter random simplicial complex as a higher dimensional extension of the classical Erdös-Rényi graph. We investigate appearance of "unusual" topological structures in the complex from the point of view of large deviations. We first study upper tail large deviation probabilities for subcomplex counts, deriving the order of magnitude of such probabilities at the logarithmic scale precision. The obtained results are then applied to analyze large deviations for the number of simplices at the critical dimension and below. Finally, these results are also used to deduce large deviation estimates for Betti numbers of the complex in the critical dimension.

math.PR↗

arXiv2021

Convergence of persistence diagram in the sparse regime

The objective of this paper is to examine the asymptotic behavior of persistence diagrams associated with Čech filtration. A persistence diagram is a graphical descriptor of a topological and algebraic structure of geometric objects. We consider Čech filtration over a scaled random sample $r_n^{-1}\mathcal X_n = \{ r_n^{-1}X_1,\dots, r_n^{-1}X_n \}$, such that $r_n\to 0$ as $n\to\infty$. We treat persistence diagrams as a point process and establish their limit theorems in the sparse regime: $nr_n^d\to0$, $n\to\infty$. In this setting, we show that the asymptotics of the $k$th persistence diagram depends on the limit value of the sequence $n^{k+2}r_n^{d(k+1)}$. If $n^{k+2}r_n^{d(k+1)} \to \infty$, the scaled persistence diagram converges to a deterministic Radon measure almost surely in the vague metric. If $r_n$ decays faster so that $n^{k+2}r_n^{d(k+1)} \to c\in (0,\infty)$, the persistence diagram weakly converges to a limiting point process without normalization. Finally, if $n^{k+2}r_n^{d(k+1)} \to 0$, the sequence of probability distributions of a persistence diagram should be normalized, and the resulting convergence will be treated in terms of the $\mathcal M_0$-topology.

math.PR↗

arXiv2021

Functional strong law of large numbers for Betti numbers in the tail

The objective of this paper is to investigate the layered structure of topological complexity in the tail of a probability distribution. We establish the functional strong law of large numbers for Betti numbers, a basic quantifier of algebraic topology, of a geometric complex outside an open ball of radius $R_n$, such that $R_n\to\infty$ as the sample size $n$ increases. The nature of the obtained law of large numbers is determined by the decay rate of a probability density. It especially depends on whether the tail of a density decays at a regularly varying rate or an exponentially decaying rate. The nature of the limit theorem depends also on how rapidly $R_n$ diverges. In particular, if $R_n$ diverges sufficiently slowly, the limiting function in the law of large numbers is crucially affected by the emergence of arbitrarily large connected components supporting topological cycles in the limit.

math.PR↗

arXiv2021

Functional strong laws of large numbers for Euler characteristic processes of extreme sample clouds

To recover the topology of a manifold in the presence of heavy tailed or exponentially decaying noise, one must understand the behavior of geometric complexes whose points lie in the tail of these noise distributions. This study advances this line of inquiry, and demonstrates functional strong laws of large numbers for the Euler characteristic process of random geometric complexes formed by random points outside of an expanding ball in $\mathbb{R}^d$. When the points are drawn from a heavy tailed distribution with a regularly varying tail, the Euler characteristic process grows at a regularly varying rate, and the scaled process converges uniformly and almost surely to a smooth function. When the points are drawn from a distribution with an exponentially decaying tail, the Euler characteristic process grows logarithmically, and the scaled process converges to another smooth function in the same sense. All of the limit theorems take place when the points inside the expanding ball are densely distributed, so that the simplex counts outside of the ball of all dimensions contribute to the Euler characteristic process.

math.PR↗

arXiv2021

Limit theorems for topological invariants of the dynamic multi-parameter simplicial complex

Topological study of existing random simplicial complexes is non-trivial and has led to several seminal works. However, the applicability of such studies is limited since the randomness there is usually governed by a single parameter. With this in mind, we focus here on the topology of the recently proposed multi-parameter random simplicial complex and, more importantly, of its dynamic analogue that we introduce here. In this dynamic setup, the temporal evolution of simplices is determined by stationary and possibly non-Markovian processes with a renewal structure. The dynamic versions of the clique complex and the Linial-Meshulum complex are special cases of our setup. Our key result concerns the regime where face-counts of a particular dimension dominate. We show that the Betti numbers corresponding to this dimension and the Euler characteristic satisfy functional strong law of large numbers and functional central limit theorems. Surprisingly, in the latter result, the limiting Gaussian process depends only upon the dynamics in the smallest non-trivial dimension.

math.PR↗

arXiv2020

A functional non-central limit theorem for multiple-stable processes with long-range dependence

A functional limit theorem is established for the partial-sum process of a class of stationary sequences which exhibit both heavy tails and long-range dependence. The stationary sequence is constructed using multiple stochastic integrals with heavy-tailed marginal distribution. Furthermore, the multiple stochastic integrals are built upon a large family of dynamical systems that are ergodic and conservative, leading to the long-range dependence phenomenon of the model. The limits constitute a new class of self-similar processes with stationary increments. They are represented by multiple stable integrals, where the integrands involve the local times of intersections of independent stationary stable regenerative sets. The joint moments of the local times are computed, which play the key in the proof and are also of independent interest.

math.PR↗

arXiv2020

Functional limit theorems for the Euler characteristic process in the critical regime

This study presents functional limit theorems for the Euler characteristic of Vietoris-Rips complexes. The points are drawn from a non-homogeneous Poisson process on $\mathbb{R}^d$, and the connectivity radius governing the formation of simplices is taken as a function of time parameter $t$, which allows us to treat the Euler characteristic as a stochastic process. The setting in which this takes place is that of the critical regime, in which the simplicial complexes are highly connected and have non-trivial topology. We establish two "functional-level" limit theorems, a strong law of large numbers and a central limit theorem for the appropriately normalized Euler characteristic process.

math.PR↗

arXiv2018

Convergence of Persistence Diagrams for Topological Crackle

In this paper we study the persistent homology associated with topological crackle generated by distributions with an unbounded support. Persistent homology is a topological and algebraic structure that tracks the creation and destruction of homological cycles (generalizations of loops or holes) in different dimensions. Topological crackle is a term that refers to homological cycles generated by "noisy" samples where the support is unbounded. We aim to establish weak convergence results for persistence diagrams - a point process representation for persistent homology, where each homological cycle is represented by its (birth,death) coordinates. In this work we treat persistence diagrams as random closed sets, so that the resulting weak convergence is defined in terms of the Fell topology. In this framework we show that the limiting persistence diagrams can be divided into two parts. The first part is a deterministic limit containing a densely-growing number of persistence pairs with a short lifespan. The second part is a two-dimensional Poisson process, representing persistence pairs with a longer lifespan.

math.PR↗