arXiv subjects
Publications and source records attributed to Louis H. Y. Chen.
In this paper, we use Stein's method to obtain optimal bounds, both in Kolmogorov and in Wasserstein distance, in the normal approximation for the empirical distribution of the ground state of a many-interacting-worlds harmonic oscillator proposed by Hall, Deckert, and Wiseman [Phys. Rev. X. (2014)]. Our bounds on the Wasserstein distance solve a conjecture of McKeague and Levin [Ann. Appl. Probab. (2016)].
We present an assessment of the distance in total variation of \textit{arbitrary} collection of prime factor multiplicities of a random number in $[n]=\{1,\dots, n\}$ and a collection of independent geometric random variables. More precisely, we impose mild conditions on the probability law of the random sample and the aforementioned collection of prime multiplicities, for which a fast decaying bound on the distance towards a tuple of geometric variables holds. Our results generalize and complement those from Kubilius et al. which consider the particular case of uniform samples in $[n]$ and collection of "small primes". As applications, we show a generalized version of the celebrated Erdös Kac theorem for not necessarily uniform samples of numbers.
This paper is a short exposition of Stein's method of normal approximation from my personal perspective. It focuses mainly on the characterization of the normal distribution and the construction of Stein identities. Through examples, it provides glimpses into the many approaches to constructing Stein identities and the diverse applications of Stein's method to mathematical problems. It also includes anecdotes of historical interest, including how Stein discovered his method and how I found an unpublished proof of his of the Berry-Esseen theorem.
We present a new perspective of assessing the rates of convergence to the Gaussian and Poisson distributions in the Erdös-Kac theorem for additive arithmetic functions $ψ$ of a random integer $J_n$ uniformly distributed over $\{1,...,n\}$. Our approach is probabilistic, working directly on spaces of random variables without any use of Fourier analytic methods, and our $ψ$ is more general than those considered in the literature. Our main results are (i) bounds on the Kolmogorov distance and Wasserstein distance between the distribution of the normalized $ψ(J_n)$ and the standard Gaussian distribution, and (ii) bounds on the Kolmogorov distance and total variation distance between the distribution of $ψ(J_n)$ and a Poisson distribution under mild additional assumptions on $ψ$. Our results generalize the existing ones in the literature.
We establish a general Berry-Esseen type bound which gives optimal bounds in many situations under suitable moment assumptions. By combining the general bound with Palm theory, we deduce a new error bound for assessing the accuracy of normal approximation to statistics arising from random measures, including stochastic geometry. We illustrate the use of the bound in four examples: completely random measures, excursion random measure of a locally dependent random process, and the total edge length of Ginibre-Voronoi tessellations and of Poisson-Voronoi tessellations. Moreover, we apply the general bound to Stein couplings and discuss the special cases of local dependence and additive functionals in occupancy problems.
In this paper, we obtain uniform and non-uniform bounds on the Kolmogorov distance in the normal approximation for Jack deformations of the character ratio, by using Stein's method and zero-bias couplings. Our uniform bound comes very close to that conjectured by Fulman [J. Combin. Theory Ser. A, 108 (2004), 275--296]. As a by-product of the proof of the non-uniform bound, we obtain a Rosenthal-type inequality for zero-bias couplings.
Applying an inductive technique for Stein and zero bias couplings yields Berry-Esseen theorems for normal approximation for two new examples. The conditions of the main results do not require that the couplings be bounded. Our two applications, one to the Erdős-Rényi, random graph with a fixed number of edges, and one to Jack measure on tableaux, demonstrate that the method can handle non-bounded variables with non-trivial global dependence, and can produce bounds in the Kolmogorov metric with the optimal rate.
In this paper we establish a framework for normal approximation for white noise functionals by Stein's method and Hida calculus. Our work is inspired by that of Nourdin and Peccati (Probab. Theory Relat. Fields 145, 75-118, 2009), who combined Stein's method and Malliavin calculus for normal approximation for functionals of Gaussian processes.
The concentration inequality approach for normal approximation by Stein's method is generalized to the multivariate setting. We use this approach to prove a non-smooth function distance for multivariate normal approximation for standardized sums of $k$-dimensional independent random vectors $W=\sum_{i=1}^n X_i$ with an error bound of order $k^{1/2}γ$ where $γ=\sum_{i=1}^n E|X_i|^3$. For sums of locally dependent (unbounded) random vectors, we obtain a fourth moment bound which is typically of order $O_k(1/\sqrt{n})$, as well as a third moment bound which is typically of order $O_k(\log n/\sqrt{n})$.
Stein's method is a method of probability approximation which hinges on the solution of a functional equation. For normal approximation the functional equation is a first order differential equation. Malliavin calculus is an infinite-dimensional differential calculus whose operators act on functionals of general Gaussian processes. Nourdin and Peccati (2009) established a fundamental connection between Stein's method for normal approximation and Malliavin calculus through integration by parts. This connection is exploited to obtain error bounds in total variation in central limit theorems for functionals of general Gaussian processes. Of particular interest is the fourth moment theorem which provides error bounds of the order $\sqrt{\mathbb{E}(F_n^4)-3}$ in the central limit theorem for elements $\{F_n\}_{n\ge 1}$ of Wiener chaos of any fixed order such that $\mathbb{E}(F_n^2) = 1$. This paper is an exposition of the work of Nourdin and Peccati with a brief introduction to Stein's method and Malliavin calculus. It is based on a lecture delivered at the Annual Meeting of the Vietnam Institute for Advanced Study in Mathematics in July 2014.
Let $\mathbb{X}=\{X_{ij}: 1\le i,j\le n\}$ be an $n\times n$ array of independent random variables where $n\ge2$. Let $π$ be a uniform random permutation of $\{1,2,\dots,n\}$, independent of $\mathbb{X}$, and let $W=\sum_{i=1}^nX_{iπ(i)}$. Suppose $\mathbb{X}$ is standardized so that ${\mathbb{E}}W=0,\operatorname {Var}(W)=1$. We prove that the Kolmogorov distance between the distribution of $W$ and the standard normal distribution is bounded by $451\sum_{i,j=1}^n{\mathbb{E}}|X_{ij}|^3/n$. Our approach is by Stein's method of exchangeable pairs and the use of a concentration inequality.
The fourth moment theorem provides error bounds of the order $\sqrt{{\mathbb E}(F^4) - 3}$ in the central limit theorem for elements $F$ of Wiener chaos of any order such that ${\mathbb E}(F^2) = 1$. It was proved by Nourdin and Peccati (2009) using Stein's method and the Malliavin calculus. It was also proved by Azmoodeh, Campese and Poly (2014) using Stein's method and Dirichlet forms. This paper is an exposition on the connections between Stein's method and the Malliavin calculus and between Stein's method and Dirichlet forms, and on how these connections are exploited in proving the fourth moment theorem.
The paper presents a general introduction to the astonishing method for deriving probability approximations that was invented by Charles Stein around 50 years ago.
We review some probabilistic properties of the sum-of-digits function of random integers. New asymptotic approximations to the total variation distance and its refinements are also derived. Four different approaches are used: a classical probability approach, Stein's method, an analytic approach and a new approach based on Krawtchouk polynomials and the Parseval identity. We also extend the study to a simple, general numeration system for which similar approximation theorems are derived.
In this paper we give a historical account of the development of Poisson approximation using Stein's method and present some of the main results. We give two recent applications, one on maximal arithmetic progressions and the other on bootstrap percolation. We also discuss generalisations to compound Poisson approximation, Poisson process approximation and multivariate Poisson approximation, and state a few open problems.
Poisson approximation using Stein's method has been extensively studied in the literature. The main focus has been on bounding the total variation distance. This paper is a first attempt on moderate deviations in Poisson approximation for right-tail probabilities of sums of dependent indicators. We obtain results under certain general conditions for local dependence as well as for size-bias coupling. These results are then applied to independent indicators, 2-runs, and the matching problem.
Stein's method is applied to obtain a general Cramer-type moderate deviation result for dependent random variables whose dependence is defined in terms of a Stein identity. A corollary for zero-bias coupling is deduced. The result is also applied to a combinatorial central limit theorem, a general system of binary codes, the anti-voter model on a complete graph, and the Curie-Weiss model. A general moderate deviation result for independent random variables is also proved.
Although the study of weak convergence of superpositions of point processes to the Poisson process dates back to the work of Grigelionis in 1963, it was only recently that Schuhmacher [Stochastic Process. Appl. 115 (2005) 1819--1837] obtained error bounds for the weak convergence. Schuhmacher considered dependent superposition, truncated the individual point processes to 0--1 point processes and then applied Stein's method to the latter. In this paper, we adopt a different approach to the problem by using Palm theory and Stein's method, thereby expressing the error bounds in terms of the mean measures of the individual point processes, which is not possible with Schuhmacher's approach. We consider locally dependent superposition as a generalization of the locally dependent point process introduced in Chen and Xia [Ann. Probab. 32 (2004) 2545--2569] and apply the main theorem to the superposition of thinned point processes and of renewal processes.