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It is well known that each statistic in the family of power divergence statistics, across n trials and r classifications with index parameter
$\lambda\in\mathbb{R}$
(the Pearson, likelihood ratio, and Freeman–Tukey statistics correspond to
$\lambda=1,0,-1/2$
, respectively), is asymptotically chi-square distributed as the sample size tends to infinity. We obtain explicit bounds on this distributional approximation, measured using smooth test functions, that hold for a given finite sample n, and all index parameters (
$\lambda>-1$
) for which such finite-sample bounds are meaningful. We obtain bounds that are of the optimal order
$n^{-1}$
. The dependence of our bounds on the index parameter
$\lambda$
and the cell classification probabilities is also optimal, and the dependence on the number of cells is also respectable. Our bounds generalise, complement, and improve on recent results from the literature.
We prove polynomial ergodicity for the one-dimensional Zig-Zag process on heavy-tailed targets and identify the exact order of polynomial convergence of the process when targeting Student distributions.
$U{\hbox{-}}\textrm{max}$
statistics were introduced by Lao and Mayer in 2008. Such statistics are natural in stochastic geometry. Examples are the maximal perimeters and areas of polygons and polyhedra formed by random points on a circle, ellipse, etc. The main method to study limit theorems for
$U{\hbox{-}}\textrm{max}$
statistics is via a Poisson approximation. In this paper we consider a general class of kernels defined on a circle, and we prove a universal limit theorem with the Weibull distribution as a limit. Its parameters depend on the degree of the kernel, the structure of its points of maximum, and the Hessians of the kernel at these points. Almost all limit theorems known so far may be obtained as simple special cases of our general theorem. We also consider several new examples. Moreover, we consider not only the uniform distribution of points but also almost arbitrary distribution on a circle satisfying mild additional conditions.
Consider a finite or infinite collection of urns, each with capacity r, and balls randomly distributed among them. An overflow is the number of balls that are assigned to urns that already contain r balls. When
$r=1$
, this is the number of balls landing in non-empty urns, which has been studied in the past. Our aim here is to use martingale methods to study the asymptotics of the overflow in the general situation, i.e. for arbitrary r. In particular, we provide sufficient conditions for both Poissonian and normal asymptotics.
Let
$X_1,X_2, \ldots, X_n$
be a sequence of independent random points in
$\mathbb{R}^d$
with common Lebesgue density f. Under some conditions on f, we obtain a Poisson limit theorem, as
$n \to \infty$
, for the number of large probability kth-nearest neighbor balls of
$X_1,\ldots, X_n$
. Our result generalizes Theorem 2.2 of [11], which refers to the special case
$k=1$
. Our proof is completely different since it employs the Chen–Stein method instead of the method of moments. Moreover, we obtain a rate of convergence for the Poisson approximation.
In this work the
$\ell_q$
-norms of points chosen uniformly at random in a centered regular simplex in high dimensions are studied. Berry–Esseen bounds in the regime
$1\leq q < \infty$
are derived and complemented by a non-central limit theorem together with moderate and large deviations in the case where
$q=\infty$
. An application to the intersection volume of a regular simplex with an
$\ell_p^n$
-ball is also carried out.
Bifurcating Markov chains (BMCs) are Markov chains indexed by a full binary tree representing the evolution of a trait along a population where each individual has two children. We provide a central limit theorem for additive functionals of BMCs under
$L^2$
-ergodic conditions with three different regimes. This completes the pointwise approach developed in a previous work. As an application, we study the elementary case of a symmetric bifurcating autoregressive process, which justifies the nontrivial hypothesis considered on the kernel transition of the BMCs. We illustrate in this example the phase transition observed in the fluctuations.
The hyperbolic random geometric graph was introduced by Krioukov et al. (Phys. Rev. E82, 2010). Among many equivalent models for the hyperbolic space, we study the d-dimensional Poincaré ball (
$d\ge 2$
), with a general connectivity radius. While many phase transitions are known for the expectation asymptotics of certain subgraph counts, very little is known about the second-order results. Two of the distinguishing characteristics of geometric graphs on the hyperbolic space are the presence of tree-like hierarchical structures and the power-law behaviour of the degree distribution. We aim to reveal such characteristics in detail by investigating the behaviour of sub-tree counts. We show multiple phase transitions for expectation and variance in the resulting hyperbolic geometric graph. In particular, the expectation and variance of the sub-tree counts exhibit an intricate dependence on the degree sequence of the tree under consideration. Additionally, unlike the thermodynamic regime of the Euclidean random geometric graph, the expectation and variance may exhibit different growth rates, which is indicative of power-law behaviour. Finally, we also prove a normal approximation for sub-tree counts using the Malliavin–Stein method of Last et al. (Prob. Theory Relat. Fields165, 2016), along with the Palm calculus for Poisson point processes.
Taylor’s power law (or fluctuation scaling) states that on comparable populations, the variance of each sample is approximately proportional to a power of the mean of the population. The law has been shown to hold by empirical observations in a broad class of disciplines including demography, biology, economics, physics, and mathematics. In particular, it has been observed in problems involving population dynamics, market trading, thermodynamics, and number theory. In applications, many authors consider panel data in order to obtain laws of large numbers. Essentially, we aim to consider ergodic behaviors without independence. We restrict our study to stationary time series, and develop different Taylor exponents in this setting. From a theoretical point of view, there has been a growing interest in the study of the behavior of such a phenomenon. Most of these works focused on the so-called static Taylor’s law related to independent samples. In this paper we introduce a dynamic Taylor’s law for dependent samples using self-normalized expressions involving Bernstein blocks. A central limit theorem (CLT) is proved under either weak dependence or strong mixing assumptions for the marginal process. The limit behavior of the estimation involves a series of covariances, unlike the classic framework where the limit behavior involves the marginal variance. We also provide an asymptotic result for a goodness-of-fit procedure suitable for checking whether the corresponding dynamic Taylor’s law holds in empirical studies.
A continuous-state branching process with immigration having branching mechanism
$\Psi$
and immigration mechanism
$\Phi$
, a CBI
$(\Psi,\Phi)$
process for short, may have either of two different asymptotic regimes, depending on whether
$\int_{0}\frac{\Phi(u)}{|\Psi(u)|}\textrm{d} u<\infty$
or
$\int_{0}\frac{\Phi(u)}{|\Psi(u)|}\textrm{d} u=\infty$
. When
$\int_{0}\frac{\Phi(u)}{|\Psi(u)|}\textrm{d} u<\infty$
, the CBI process has either a limit distribution or a growth rate dictated by the branching dynamics. When
$\scriptstyle\int_{0}\tfrac{\Phi(u)}{|\Psi(u)|}\textrm{d} u=\infty$
, immigration overwhelms branching dynamics. Asymptotics in the latter case are studied via a nonlinear time-dependent renormalization in law. Three regimes of weak convergence are exhibited. Processes with critical branching mechanisms subject to a regular variation assumption are studied. This article proves and extends results stated by M. Pinsky in ‘Limit theorems for continuous state branching processes with immigration’ (Bull. Amer. Math. Soc.78, 1972).
The generalized perturbative approach is an all-purpose variant of Stein’s method used to obtain rates of normal approximation. Originally developed for functions of independent random variables, this method is here extended to functions of the realization of a hidden Markov model. In this dependent setting, rates of convergence are provided in some applications, leading, in each instance, to an extra log-factor vis-à-vis the rate in the independent case.
The random-cluster model is a unifying framework for studying random graphs, spin systems and electrical networks that plays a fundamental role in designing efficient Markov Chain Monte Carlo (MCMC) sampling algorithms for the classical ferromagnetic Ising and Potts models. In this paper, we study a natural non-local Markov chain known as the Chayes–Machta (CM) dynamics for the mean-field case of the random-cluster model, where the underlying graph is the complete graph on n vertices. The random-cluster model is parametrised by an edge probability p and a cluster weight q. Our focus is on the critical regime:
$p = p_c(q)$
and
$q \in (1,2)$
, where
$p_c(q)$
is the threshold corresponding to the order–disorder phase transition of the model. We show that the mixing time of the CM dynamics is
$O({\log}\ n \cdot \log \log n)$
in this parameter regime, which reveals that the dynamics does not undergo an exponential slowdown at criticality, a surprising fact that had been predicted (but not proved) by statistical physicists. This also provides a nearly optimal bound (up to the
$\log\log n$
factor) for the mixing time of the mean-field CM dynamics in the only regime of parameters where no non-trivial bound was previously known. Our proof consists of a multi-phased coupling argument that combines several key ingredients, including a new local limit theorem, a precise bound on the maximum of symmetric random walks with varying step sizes and tailored estimates for critical random graphs. In addition, we derive an improved comparison inequality between the mixing time of the CM dynamics and that of the local Glauber dynamics on general graphs; this results in better mixing time bounds for the local dynamics in the mean-field setting.
For a uniform random labelled tree, we find the limiting distribution of tree parameters which are stable (in some sense) with respect to local perturbations of the tree structure. The proof is based on the martingale central limit theorem and the Aldous–Broder algorithm. In particular, our general result implies the asymptotic normality of the number of occurrences of any given small pattern and the asymptotic log-normality of the number of automorphisms.
Let
$(Z_n)_{n\geq 0}$
be a critical branching process in a random environment defined by a Markov chain
$(X_n)_{n\geq 0}$
with values in a finite state space
$\mathbb{X}$
. Let
$ S_n = \sum_{k=1}^n \ln f_{X_k}^{\prime}(1)$
be the Markov walk associated to
$(X_n)_{n\geq 0}$
, where
$f_i$
is the offspring generating function when the environment is
$i \in \mathbb{X}$
. Conditioned on the event
$\{ Z_n>0\}$
, we show the nondegeneracy of the limit law of the normalized number of particles
${Z_n}/{e^{S_n}}$
and determine the limit of the law of
$\frac{S_n}{\sqrt{n}} $
jointly with
$X_n$
. Based on these results we establish a Yaglom-type theorem which specifies the limit of the joint law of
$ \log Z_n$
and
$X_n$
given
$Z_n>0$
.
We establish a normal approximation for the limiting distribution of partial sums of random Rademacher multiplicative functions over function fields, provided the number of irreducible factors of the polynomials is small enough. This parallels work of Harper for random Rademacher multiplicative functions over the integers.
We revisit the so-called cat-and-mouse Markov chain, studied earlier by Litvak and Robert (2012). This is a two-dimensional Markov chain on the lattice
$\mathbb{Z}^2$
, where the first component (the cat) is a simple random walk and the second component (the mouse) changes when the components meet. We obtain new results for two generalisations of the model. First, in the two-dimensional case we consider far more general jump distributions for the components and obtain a scaling limit for the second component. When we let the first component be a simple random walk again, we further generalise the jump distribution of the second component. Secondly, we consider chains of three and more dimensions, where we investigate structural properties of the model and find a limiting law for the last component.
For a one-locus haploid infinite population with discrete generations, the celebrated model of Kingman describes the evolution of fitness distributions under the competition of selection and mutation, with a constant mutation probability. This paper generalises Kingman’s model by using independent and identically distributed random mutation probabilities, to reflect the influence of a random environment. The weak convergence of fitness distributions to the globally stable equilibrium is proved. Condensation occurs when almost surely a positive proportion of the population travels to and condenses at the largest fitness value. Condensation may occur when selection is favoured over mutation. A criterion for the occurrence of condensation is given.
We give a setting of the Diaconis–Freedman chain in a multi-dimensional simplex and consider its asymptotic behavior. By using techniques from random iterated function theory and quasi-compact operator theory, we first give some sufficient conditions which ensure the existence and uniqueness of an invariant probability measure and, in particular cases, explicit formulas for the invariant probability density. Moreover, we completely classify all behaviors of this chain in dimension two. Some other settings of the chain are also discussed.
We investigate vertex levels of containment in a random hypergraph grown in the spirit of a recursive tree. We consider a local profile tracking the evolution of the containment of a particular vertex over time, and a global profile concerned with counts of the number of vertices of a particular containment level.
For the local containment profile, we obtain the exact mean, variance, and probability distribution in terms of standard combinatorial quantities such as generalized harmonic numbers and Stirling numbers of the first kind. Asymptotically, we observe phases: the early vertices have an asymptotically normal distribution, intermediate vertices have a Poisson distribution, and late vertices have a degenerate distribution.
As for the global containment profile, we establish an asymptotically normal distribution for the number of vertices at the smallest containment level as well as their covariances with the number of vertices at the second smallest containment level and the variances of these numbers. The orders in the variance–covariance matrix establish concentration laws.