Stein's Method Research Articles

Estimating intensity functions of spatial inhomogeneous Poisson point processes via a Stein estimator

We develop a global Stein estimator through Malliavin derivatives to estimate the multi-variable intensity functions of general d-dimensional spatial inhomogeneous Poisson point processes (IPPPs). The gain performance between our Stein estimator and a tested maximum likelihood (ML) estimator is analyzed via Malliavin calculus. In comparison with existing homogeneous studies, our inhomogeneous cases are more complex in a global way and more real in practice. In the numerical simulation experiments, our estimated intensity functions via the Stein estimator well match the true one, which outperform the ones obtained by the ML estimator with an average gain larger than 30% concerning mean squared errors (MSEs). Hence, our Stein method is more robust with smaller MSEs. We also establish the mean-squared convergence with error estimation for our Stein estimator. Comparing with the existing study, our observation window for an IPPP is a d-dimensional rectangle and its boundary is not -smooth. This non-smoothness introduces additional complexity to our convergence analysis. Thus, a weak convergence approach is employed to overcome this difficulty.

Bounds for an integral of the modified Bessel function of the first kind and expressions involving it

Simple upper and lower bounds are obtained for the integral ∫0xe−γttνIν(t)dt, x>0, ν>−12, 0<γ<1. Most of our bounds for this integral are tight as x→∞. We apply one of our inequalities to bound some expressions involving this integral. Two of these expressions appear in Stein's method for variance-gamma approximation, and our bounds will allow for a technical advancement to be made to the method.

Steady‐state analysis of load balancing with Coxian‐2 distributed service times

AbstractThis paper studies load balancing for many‐server (N servers) systems. Each server has a buffer of size b − 1, and can have at most one job in service and b − 1 jobs in the buffer. The service time of a job follows the Coxian‐2 distribution. We focus on steady‐state performance of load balancing policies in the heavy traffic regime such that the normalized load of system is λ = 1 − N−α for 0 &lt; α &lt; 0.5. We identify a set of policies that achieve asymptotic zero waiting. The set of policies include several classical policies such as join‐the‐shortest‐queue (JSQ), join‐the‐idle‐queue (JIQ), idle‐one‐first (I1F) and power‐of‐d‐choices (Po d) with d = O(Nα log N). The proof of the main result is based on Stein's method and state space collapse. A key technical contribution of this paper is the iterative state space collapse approach that leads to a simple generator approximation when applying Stein's method.

KOLMOGOROV BOUNDS FOR THE NORMAL APPROXIMATION OF THE NUMBER OF TRIANGLES IN THE ERDŐS–RÉNYI RANDOM GRAPH

We bound the error for the normal approximation of the number of triangles in the Erdős–Rényi random graph with respect to the Kolmogorov metric. Our bounds match the best available Wasserstein bounds obtained by Barbouret al.[(1989). A central limit theorem for decomposable random variables with applications to random graphs.Journal of Combinatorial Theory, Series B47: 125–145], resolving a long-standing open problem. The proofs are based on a new variant of the Stein–Tikhomirov method—a combination of Stein's method and characteristic functions introduced by Tikhomirov [(1976). The rate of convergence in the central limit theorem for weakly dependent variables.Vestnik Leningradskogo Universiteta158–159, 166].

Stein's method, heat kernel, and linear functions on the orthogonal groups

Combining Stein's method with heat kernel techniques, we study the function Tr(AO), where A is a fixed n×n matrix over R such that Tr(AAt)=n, and O is from the Haar measure of the orthogonal group O(n,R). It is shown that the total variation distance of the random variable Tr(AO) to a standard normal random variable is bounded by 22n−1, slightly improving the constant in a bound of Meckes, which was obtained by completely different methods.

Cramér-type moderate deviation theorems for nonnormal approximation

A Cram\'er-type moderate deviation theorem quantifies the relative error of the tail probability approximation. It provides theoretical justification when the limiting tail probability can be used to estimate the tail probability under study. Chen Fang and Shao (2013) obtained a general Cram\'er-type moderate result using Stein's method when the limiting was a normal distribution. In this paper, Cram\'er-type moderate deviation theorems are established for nonnormal approximation under a general Stein identity, which is satisfied via the exchangeable pair approach and Stein's coupling. In particular, a Cram\'er-type moderate deviation theorem is obtained for the general Curie--Weiss model and the imitative monomer-dimer mean-field model.

On the Order of Approximation in Limit Theorems for Negative–Binomial Sums of Strictly Stationary m-Dependent Random Variables

During the last several decades, the results related to geometric random sums of independent identically distributed random variables have become interesting results in probability theory and in insurance, risk theory, queuing theory and stochastic finance, etc. The negative–binomial random sums are extensions of geometric random sums. Up to the present, the negative–binomial random sums of independent identically distributed random variables have attracted much attention since they appear in many fields. However, for the strictly stationary m-dependent summands, such negative–binomial random sums are rarely studied. It seems that the complexity of the dependent structure has limited the use of classical tools based on independent random variables such as characteristic function. Up to now, a number of methods have been used to overcome the dependency such as the truncate method, Stein method, and Heinrich’s method. The paper deals with the order of approximation in weak limit theorems for normalized negative–binomial sums of strictly stationary m-dependent random variables generated by a sequence of independent, identically distributed random variables, using the moving average techniques. The orders of approximation in desired theorems are established in terms of the so-called Zolotarev ζ-metric. The obtained results are extensions and generalizations of several known results related to negative–binomial random sums and geometric random sums of independent, identically distributed random variables.

Stein's method in two limit theorems involving the generalized inverse Gaussian distribution

The generalized hyperbolic (GH) distribution converges in law to the generalized inverse Gaussian (GIG) distribution under certain conditions on the parameters. When the edges of an infinite rooted tree are equipped with independent resistances that are inverse Gaussian or reciprocal inverse Gaussian distributions, the total resistance converges almost surely to some random variable which follows the reciprocal inverse Gaussian (RIG) distribution. In this paper we provide explicit upper bounds for the distributional distance between GH (resp. infinite tree) distribution and their limiting GIG (resp. RIG) distribution applying Stein's method.

Poisson approximation and connectivity in a scale-free random connection model

We study an inhomogeneous random connection model in the connectivity regime. The vertex set of the graph is a homogeneous Poisson point process $\mathcal{P}_s$ of intensity $s>0$ on the unit cube $S=\left(-\frac{1}{2},\frac{1}{2}\right]^{d},$ $d \geq 2$ . Each vertex is endowed with an independent random weight distributed as $W$, where $P(W>w)=w^{-\beta}1_{[1,\infty)}(w)$, $\beta>0$. Given the vertex set and the weights an edge exists between $x,y\in \mathcal{P}_s$ with probability $\left(1 - \exp\left( - \frac{\eta W_xW_y}{\left(d(x,y)/r\right)^{\alpha}} \right)\right),$ independent of everything else, where $\eta, \alpha > 0$, $d(\cdot, \cdot)$ is the toroidal metric on $S$ and $r > 0$ is a scaling parameter. We derive conditions on $\alpha, \beta$ such that under the scaling $r_s(\xi)^d= \frac{1}{c_0 s} \left( \log s +(k-1) \log\log s +\xi+\log\left(\frac{\alpha\beta}{k!d} \right)\right),$ $\xi \in \mathbb{R}$, the number of vertices of degree $k$ converges in total variation distance to a Poisson random variable with mean $e^{-\xi}$ as $s \to \infty$, where $c_0$ is an explicitly specified constant that depends on $\alpha, \beta, d$ and $\eta$ but not on $k$. In particular, for $k=0$ we obtain the regime in which the number of isolated nodes stabilizes, a precursor to establishing a threshold for connectivity. We also derive a sufficient condition for the graph to be connected with high probability for large $s$. The Poisson approximation result is derived using the Stein's method.

Stein's method in two limit theorems involving the generalized inverse Gaussian distribution

Sous certaines conditions sur ses parametres, la loi hyperbolique generalisee (GH) converge vers la loi gaussienne inverse generalisee (GIG). Lorsque les aretes d'un arbre infini sont munies de resistances aleatoires independantes, de loi gaussienne inverse ou de loi gaussienne inverse reciproque, la resistance equivalente converge presque surement vers une variable aleatoire de loi gaussienne inverse reciproque (RIG). Dans cet article, nous determinons des majorants explicites de la distance probabiliste entre la loi GH (resp. un circuit arborescent) et la loi limite GIG (resp. RIG) en appliquant la methode de Stein.

Variations on Salem–Zygmund results for random trigonometric polynomials: application to almost sure nodal asymptotics

On a probability space $(\Omega, \mathcal F, \mathbb P)$ we consider two independent sequences $(a_k)_{k \geq 1}$ and $(b_k)_{k \geq 1}$ of i.i.d. random variables that are centered with unit variance and which admit a moment strictly higher than two. We define the associated random trigonometric polynomial \[ f_n(t) :=\frac{1}{\sqrt{n}} \sum_{k=1}^n a_k \cos(kt)+b_k \sin(kt), \quad t \in \mathbb R. \] In their seminal work, for Rademacher coefficients, Salem and Zygmund showed that $\mathbb P$ almost surely: \[ \forall t\in\mathbb R,\quad \frac{1}{2\pi}\int_{0}^{2\pi} \exp\left(i t f_n(x)\right) dx \xrightarrow[n\to\infty]~e^{-\frac{t^2}{2}}. \] In other words, if $X$ denotes an independent random variable uniformly distributed on $[0,2\pi]$, $\mathbb{P}$ almost surely, under the law of $X$, $f_n(X)$ converges in distribution to a standard Gaussian variable. In this paper, we revisit the above result from different perspectives. Namely, i) we establish a possibly sharp convergence rate for some adequate metric via the Stein's method, ii) we prove a functional counterpart of Salem--Zygmund CLT, iii) we extend it to more general distributions for $X$, iv) we also prove that the convergence actually holds in total variation. As an application, in the case where the random coefficients have a symmetric distribution and admit a moment of order $4$, we show that $\mathbb{P}$ almost surely, for any interval $[a,b] \subset [0, 2\pi]$ \[\frac{\mathcal N(f_n,[a,b])}{n} \xrightarrow[n \to +\infty]{} \frac{(b-a)}{\pi \sqrt{3}},\] where $\mathcal N(f_n,[a,b])$ denotes the number of real zeros of $f_n$ in the interval $[a,b]$. To the best of our knowledge, such an almost sure result is new in the framework of random trigonometric polynomials, even in the case of Gaussian coefficients.

Malliavin–Stein method: a survey of some recent developments

Initiated around the year 2007, the Malliavin–Stein approach to probabilistic approximations combines Stein’s method with infinite-dimensional integration by parts formulae based on the use of Malliavin-type operators. In the last decade, Malliavin–Stein techniques have allowed researchers to establish new quantitative limit theorems in a variety of domains of theoretical and applied stochastic analysis. The aim of this survey is to illustrate some of the latest developments of the Malliavin–Stein method, with specific emphasis on extensions and generalizations in the framework of Markov semigroups and of random point measures.

Convergence Rate of Random Geometric Sum Distributions to the Laplace Law

In this paper we modify the Stein method and the auxiliary technique of distributional transformations of random variables. This enables us to estimate the convergence rate of distributions of norm...

Wilson Loop Expectations in Lattice Gauge Theories with Finite Gauge Groups

Wilson loop expectations at weak coupling are computed to first order, for four dimensional lattice gauge theories with finite gauge groups which satisfy some mild additional conditions. This continues recent work of Chatterjee, which considered the case of gauge group $\mathbb{Z}_2$. The main steps are (1) reducing the first order computation to a problem of Poisson approximation, and (2) using Stein's method to carry out the Poisson approximation.

Functional approximations via Stein’s method of exchangeable pairs

We combine the method of exchangeable pairs with Stein's method for functional approximation. As a result, we give a general linearity condition under which an abstract Gaussian approximation theorem for stochastic processes holds. We apply this approach to estimate the distance of a sum of random variables, chosen from an array according to a random permutation, from a Gaussian mixture process. This result lets us prove a functional combinatorial central limit theorem. We also consider a graph-valued process and bound the speed of convergence of the distribution of its rescaled edge counts to a continuous Gaussian process.

A non-increasing tree growth process for recursive trees and applications

AbstractWe introduce a non-increasing tree growth process $((T_n,{\sigma}_n),\, n\ge 1)$ , where Tn is a rooted labelled tree on n vertices and σn is a permutation of the vertex labels. The construction of (Tn, σn) from (Tn−1, σn−1) involves rewiring a random (possibly empty) subset of edges in Tn−1 towards the newly added vertex; as a consequence Tn−1 ⊄ Tn with positive probability. The key feature of the process is that the shape of Tn has the same law as that of a random recursive tree, while the degree distribution of any given vertex is not monotone in the process.We present two applications. First, while couplings between Kingman’s coalescent and random recursive trees were known for any fixed n, this new process provides a non-standard coupling of all finite Kingman’s coalescents. Second, we use the new process and the Chen–Stein method to extend the well-understood properties of degree distribution of random recursive trees to extremal-range cases. Namely, we obtain convergence rates on the number of vertices with degree at least $c\ln n$ , c ∈ (1, 2), in trees with n vertices. Further avenues of research are discussed.

Local limit theorems for occupancy models

We present a rather general method for proving local limit theorems, with a good rate of convergence, for sums of dependent random variables. The method is applicable when a Stein coupling can be exhibited. Our approach involves both Stein's method for distributional approximation and Stein's method for concentration. As applications, we prove local central limit theorems with rate of convergence for the number of germs with d neighbors in a germ‐grain model, and the number of degree‐d vertices in an Erdős‐Rényi random graph. In both cases, the error rate is optimal, up to logarithmic factors.

Sunklodas’ Approach to Normal Approximation for Time-Dependent Dynamical Systems

We consider time-dependent dynamical systems arising as sequential compositions of self-maps of a probability space. We establish conditions under which the Birkhoff sums for multivariate observations, given a centering and a general normalizing sequence $b(N)$ of invertible square matrices, are approximated by a normal distribution with respect to a metric of regular test functions. Depending on the metric and the normalizing sequence $b(N)$, the conditions imply that the error in the approximation decays either at the rate $O(N^{-1/2})$ or the rate $O(N^{-1/2} \log N)$, under the additional assumption that $\Vert b(N)^{-1} \Vert \lesssim N^{-1/2}$. The error comes with a multiplicative constant whose exact value can be computed directly from the conditions. The proof is based on an observation due to Sunklodas regarding Stein's method of normal approximation. We give applications to one-dimensional random piecewise expanding maps and to sequential, random, and quasistatic intermittent systems.

The Breuer–Major theorem in total variation: Improved rates under minimal regularity

In this paper we prove an estimate for the total variation distance, in the framework of the Breuer–Major theorem, using the Malliavin–Stein method, assuming the underlying function g to be once weakly differentiable with g and g′ having finite moments of order four with respect to the standard Gaussian density. This result is proved by a combination of Gebelein’s inequality and some novel estimates involving Malliavin operators.

Trees grown under young-age preferential attachment

AbstractWe introduce a class of non-uniform random recursive trees grown with an attachment preference for young age. Via the Chen–Stein method of Poisson approximation, we find that the outdegree of a node is characterized in the limit by ‘perturbed’ Poisson laws, and the perturbation diminishes as the node index increases. As the perturbation is attenuated, a pure Poisson limit ultimately emerges in later phases. Moreover, we derive asymptotics for the proportion of leaves and show that the limiting fraction is less than one half. Finally, we study the insertion depth in a random tree in this class. For the insertion depth, we find the exact probability distribution, involving Stirling numbers, and consequently we find the exact and asymptotic mean and variance. Under appropriate normalization, we derive a concentration law and a limiting normal distribution. Some of these results contrast with their counterparts in the uniform attachment model, and some are similar.