A note on Stein equation for weighted sums of independent $χ^{2}$ distributions

Stein's method and the Laplace distribution

Using Stein's method techniques, we develop a framework which allows one to bound the error terms arising from approximation by the Laplace distribution and apply it to the study of random sums of mean zero random variables. As a corollary, we deduce a Berry-Esseen type theorem for the convergence of certain geometric sums. Our results make use of a second order characterizing equation and a distributional transformation which is related to zero-biasing.

关于随机图极值度分布的注记：基于Stein-Chen方法

A note on the distribution of the extreme degrees of a random graph via the Stein-Chen method

本文用Stein-Chen方法重新证明Bollobás关于随机图极值度分布的定理，并给出收敛速率，且方法可推广至边概率依赖顶点数的情形。

基于Stein方法的局部相依随机变量和的三参数逼近

Three-Parameter Approximations of Sums of Locally Dependent Random Variables via Stein's Method

本文利用Stein方法为局部相依非负整数值随机变量之和建立三参数随机变量逼近的全变差距离上界，并在四个经典例子中获得比现有结果更精确的误差界。

On Strong Embeddings by Stein's Method

Strong embeddings, that is, couplings between a partial sum process of a sequence of random variables and a Brownian motion, have found numerous applications in probability and statistics. We extend Chatterjee's novel use of Stein's method for $\{-1,+1\}$ valued variables to a general class of discrete distributions, and provide $\log n$ rates for the coupling of partial sums of independent variables to a Brownian motion, and results for coupling sums of suitably standardized exchangeable variables to a Brownian bridge.

On the Entropy of Sums of Bernoulli Random Variables via the Chen-Stein Method

This paper considers the entropy of the sum of (possibly dependent and non-identically distributed) Bernoulli random variables. Upper bounds on the error that follows from an approximation of this entropy by the entropy of a Poisson random variable with the same mean are derived. The derivation of these bounds combines elements of information theory with the Chen-Stein method for Poisson approximation. The resulting bounds are easy to compute, and their applicability is exemplified. This conference paper presents in part the first half of the paper entitled "An information-theoretic perspectiv…

通过 Stein 算子扰动逼近局部相依随机变量之和

Approximation of Sums of Locally Dependent Random Variables via Perturbation of Stein Operator

本文利用 Stein 方法及算子扰动，为局部相依非负整值随机变量之和逼近 Poisson–二项混合或负二项分布建立一般总变差误差界，并将 runs 应用的收敛阶改进至 $O(|J|^{-1})$。

Multivariate Normal Approximation by Stein's Method: The Concentration Inequality Approach

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…

A Note on Stein's Method for Heavy-Traffic Analysis

In this note, we apply Stein's method to analyze the steady-state distribution of queueing systems in the traditional heavy-traffic regime. Compared to previous methods (e.g., drift method and transform method), Stein's method allows us to establish stronger results with simple and template proofs. In particular, we consider discrete-time systems in this note. We first introduce the key ideas of Stein's method for heavy-traffic analysis through a single-server system. Then, we apply the developed template to analyze both load balancing problems and scheduling problems. All these three examples…

Approximations to Weighted Sums of Random Variables

In this paper, we obtain error bound for binomial and negative binomial approximations to weighted sums of locally dependent random variables, using Stein's method. We also discuss approximation results for weighted sums of independent random variables. We demonstrate our results through some applications in finance and runs in statistics.

A note on existence of free Stein kernels

Stein kernels are a way of comparing probability distributions, defined via integration by parts formulas. We provide two constructions of Stein kernels in free probability. One is given by an explicit formula, and the other via free Poincaré inequalities. In particular, we show that unlike in the classical setting, free Stein kernels always exist. As corollaries, we derive new bounds on the rate of convergence in the free CLT, and a strengthening of a characterization of the semicircular law due to Biane.

A note on the exchangeability condition in Stein's method

We show by a surprisingly simple argument that the exchangeability condition, which is key to the exchangeable pair approach in Stein's method for distributional approximation, can be omitted in many standard settings. This is achieved by replacing the usual antisymmetric function by a simpler one, for which only equality in distribution is required. In the case of normal approximation we also slightly improve the constants appearing in previous results. For Poisson approximation, a different antisymmetric function is used, and additional error terms are needed if the bound is to be extended b…

Stein方法用于非对称拉普拉斯近似

Stein's method for asymmetric Laplace approximation

本文发展了Stein方法用于非对称拉普拉斯分布近似，推导了在Kolmogorov、Wasserstein和光滑Wasserstein距离下的一般界，并应用于几何随机和确定性随机变量之和的非对称拉普拉斯近似。

Note on A. Barbour's paper on Stein's method for diffusion approximations

In (Barbour, 1990) foundations for diffusion approximation via Stein's method are laid. This paper has been cited more than 130 times and is a cornerstone in the area of Stein's method. A semigroup argument is used therein to solve a Stein equation for Gaussian diffusion approximation. We prove that, contrary to the claim in (Barbour, 1990), the semigroup considered therein is not strongly continuous on the Banach space of continuous, real-valued functions on D[0,1] growing slower than a cubic, equipped with an appropriate norm. We also provide a proof of the exact formulation of the solution …

Mod-Poisson approximation schemes and higher-order Chen-Stein inequalities

In this article, we provide an extension of the Chen-Stein inequality for Poisson approximation in the total variation distance for sums of independent Bernoulli random variables in two ways. We prove that we can improve the rate of convergence (hence the quality of the approximation) by using explicitly constructed signed or positive probability measures, and that we can extend the setting to possibly dependent random variables. The framework which allows this is that of mod-Poisson convergence, and more precisely those mod-Poisson convergent sequences whose residue functions can be expressed…

Satterthwaite近似的显式Gamma-Stein界

Explicit Gamma-Stein Bounds for Satterthwaite's Approximation

本文为Satterthwaite近似推导显式Gamma-Stein界，通过核表示和矩匹配控制近似误差，应用于方差估计与高维统计。

局部相依随机变量加权和的 Poisson 逼近

Poisson Approximation to Weighted Sums of Locally Dependent Random Variables

该研究利用 Stein 方法分析局部相依随机变量加权和的 Poisson 逼近，给出优于已有结论的误差界，并通过数值比较验证其有效性。

Improved nonuniform Berry--Esseen-type bounds

New nonuniform Berry--Esseen-type bounds for sums of independent random variables are obtained, motivated by recent studies concerning such bounds for nonlinear statistics. The proofs are based on the Chen--Shao concentration techniques within the framework of the Stein method.

On Stein operators for discrete approximations

In this paper, a new method based on probability generating functions is used to obtain multiple Stein operators for various random variables closely related to Poisson, binomial and negative binomial distributions. Also, Stein operators for certain compound distributions, where the random summand satisfies Panjer's recurrence relation, are derived. A well-known perturbation approach for Stein's method is used to obtain total variation bounds for the distributions mentioned above. The importance of such approximations is illustrated, for example, by the binomial convoluted with Poisson approxi…

Approximation Results for Sums of Independent Random Variables

In this article, we consider Poisson and Poisson convoluted geometric approximation to the sums of $n$ independent random variables under moment conditions. We use Stein's method to derive the approximation results in total variation distance. The error bounds obtained are either comparable to or improvement over the existing bounds available in the literature. Also, we give an application to the waiting time distribution of 2-runs.

Stein approximation for functionals of independent random sequences

We derive Stein approximation bounds for functionals of uniform random variables, using chaos expansions and the Clark-Ocone representation formula combined with derivation and finite difference operators. This approach covers sums and functionals of both continuous and discrete independent random variables. For random variables admitting a continuous density, it recovers classical distance bounds based on absolute third moments, with better and explicit constants. We also apply this method to multiple stochastic integrals that can be used to represent U-statistics, and include linear and quad…

A note on the invariance principle of the product of sums of random variables

In literature, the central limit theorems for the product of sums of various random variables have studied. The purpose of this note is to show that this kind of results are corollary of the invariance principle.

Skewness correction in tail probability approximations for sums of local statistics

Correcting for skewness can result in more accurate tail probability approximations in the central limit theorem for sums of independent random variables. In this paper, we extend the theory to sums of local statistics of independent random variables and apply the result to $k$-runs, U-statistics, and subgraph counts in the Erdös-Rényi random graph. To prove our main result, we develop exponential concentration inequalities and higher-order Cramér-type moderate deviations via Stein's method.

A note on one-dimensional Poincaré inequalities by Stein-type integration

We study the weighted Poincaré constant $C(p,w)$ of a probability density $p$ with weight function $w$ using integration methods inspired by Stein's method. We obtain a new version of the Chen-Wang variational formula which, as a byproduct, yields simple upper and lower bounds on $C(p,w)$ in terms of the so-called Stein kernel of $p$. We also iterate these variational formulas so as to build sequences of nested intervals containing the Poincaré constant, sequences of functions converging to said constant, as well as sequences of functions converging to the solutions of the corresponding spectr…

Symmetric and centered binomial approximation of sums of locally dependent random variables

Stein's method is used to approximate sums of discrete and locally dependent random variables by a centered and symmetric Binomial distribution. Under appropriate smoothness properties of the summands, the same order of accuracy as in the Berry-Essen Theorem is achieved. The approximation of the total number of points of a point processes is also considered. The results are applied to the exceedances of the $r$-scans process and to the Matérn hardcore point process type I.

Dickman Approximation of weighted random sums in the Kolmogorov distance

We consider distributional approximation by generalized Dickman distributions, which appear in number theory, perpetuities, logarithmic combinatorial structures and many other areas. We prove bounds in the Kolmogorov distance for the approximation of certain weighted sums of Bernoulli and Poisson random variables by members of this family. While such results have previously been shown in Bhattacharjee and Goldstein (2019) for distances based on smoother test functions and for a special case of the random variables considered in this paper, results in the Kolmogorov distance are new. We also es…

Stein's Method for Tempered Stable Distributions

In this article, we develop Stein characterization for two-sided tempered stable distribution. Stein characterizations for normal, gamma, Laplace, and variance-gamma distributions already known in the literature follow easily. One can also derive Stein characterizations for more difficult distributions such as the distribution of product of two normal random variables, a difference between two gamma random variables. Using the semigroup approach, we obtain estimates of the solution to Stein equation. Finally, we apply these estimates to obtain error bounds in the Wasserstein-type distance for …

与三元独立随机变量相关的近似

Variance gamma approximation to sums of triplewise independent random variables

探讨三元独立随机变量与完全二分图的联系并构造其序列，用斯坦因方法对其和进行方差伽马近似并推导边界，还讨论了相关极限定理。

A note on the Erdős distinct subset sums problem

We present two short proofs giving the best known asymptotic lower bound for the maximum element in a set of $n$ positive integers with distinct subset sums.

A note on sums of independent random variables

In this note a two sided bound on the tail probability of sums of independent, and either symmetric or nonnegative, random variables is obtained. We utilize a recent result by Latała on bounds on moments of such sums. We also give a new proof of Latała's result for nonnegative random variables, and improve one of the constants in his inequality.

A note on the normal approximation error for randomly weighted self-normalized sums

Let $\bX=\{X_n\}_{n\geq 1}$ and $\bY=\{Y_n\}_{n\geq 1}$ be two independent random sequences. We obtain rates of convergence to the normal law of randomly weighted self-normalized sums $$ ψ_n(\bX,\bY)=\sum_{i=1}^nX_iY_i/V_n,\quad V_n=\sqrt{Y_1^2+...+Y_n^2}. $$ These rates are seen to hold for the convergence of a number of important statistics, such as for instance Student's $t$-statistic or the empirical correlation coefficient.

Malliavin-Stein method for Variance-Gamma approximation on Wiener space

We combine Malliavin calculus with Stein's method to derive bounds for the Variance-Gamma approximation of functionals of isonormal Gaussian processes, in particular of random variables living inside a fixed Wiener chaos induced by such a process. The bounds are presented in terms of Malliavin operators and norms of contractions. We show that a sequence of distributions of random variables in the second Wiener chaos converges to a Variance-Gamma distribution if and only if their moments of order two to six converge to that of a Variance-Gamma distributed random variable (six moment theorem). M…

Variance-Gamma approximation via Stein's method

Variance-Gamma distributions are widely used in financial modelling and contain as special cases the normal, Gamma and Laplace distributions. In this paper we extend Stein's method to this class of distributions. In particular, we obtain a Stein equation and smoothness estimates for its solution. This Stein equation has the attractive property of reducing to the known normal and Gamma Stein equations for certain parameter values. We apply these results and local couplings to bound the distance between sums of the form $\sum_{i,j,k=1}^{m,n,r}X_{ik}Y_{jk}$, where the $X_{ik}$ and $Y_{jk}$ are in…

Stein's method, semicircle distribution, and reduced decompositions of the longest element in the symmetric group

Consider a uniformly chosen random reduced decomposition of the longest element in the symmetric group. It is known that the location of the first transposition in this decomposition converges to the semicircle distribution. In this note we provide a sharp error term for this result, using the "comparison of generators" approach to Stein's method.

两等位基因Moran模型的Beta近似：Stein方法

Beta approximation for the two alleles Moran model by Stein's method

本文针对两等位基因Moran模型，利用Stein方法给出Beta近似的精确误差项，并证明其符合Doebler框架。

A note on limiting behaviour of constrained sums of two variables

This note studies the asymptotic properties of the variable $$Z_d:=\frac{X_1}{d}|\{X_1+X_2=d\},$$ as $d\to \infty$. Here $X_1$ and $X_2$ are non-negative i.i.d. variables with a common twice differentiable density function $f$. General results concerning the distributional limits of $Z_d$ are discussed with various examples. Eventual log-convexity or log-concavity of $f$ turns out to be the key ingredient that determines how the variable $Z_d$ behaves. As a consequence, two surprising discoveries are presented: Firstly, it is noted that the distributional limit is not strictly determined by th…

Geometric sums, size biasing and zero biasing

The geometric sum plays a significant role in risk theory and reliability theory \cite{Kala97} and a prototypical example of the geometric sum is Rényi's theorem~\cite{Renyi56} saying a sequence of suitably parameterised geometric sums converges to the exponential distribution. There is extensive study of the accuracy of exponential distribution approximation to the geometric sum \cite{Sugakova95,Kala97,PekozRollin11} but there is little study on its natural counterpart of gamma distribution approximation to negative binomial sums. In this note, we show that a nonnegative random variable follo…

关于伯努利矩阵随机对角和的泊松逼近

On the Poisson approximation of random diagonal sums of Bernoulli matrices

本文利用 Stein-Chen 方法建立伯努利矩阵随机对角和与泊松分布之间全变差、Wasserstein 及局部距离的显式不等式，并改进经典泊松逼近定理 4.A 的部分界。

Stein operators for variables form the third and fourth Wiener chaoses

Let $Z$ be a standard normal random variable and let $H_n$ denote the $n$-th Hermite polynomial. In this note, we obtain Stein equations for the random variables $H_3(Z)$ and $H_4(Z)$, which represents a first step towards developing Stein's method for distributional limits from the third and fourth Wiener chaoses. Perhaps surprisingly, these Stein equations are fifth and third order linear ordinary differential equations, respectively. As a warm up, we obtain a Stein equation for the random variable $aZ^2+bZ+c$, $a,b,c\in\mathbb{R}$, which leads us to a Stein equation for the non-central chi-…

具有大素因子的随机可乘函数部分和的分布

The distribution of partial sums of random multiplicative functions with a large prime factor

本文研究 Steinhaus 随机可乘函数在限制最大素因子大于 $\sqrt{x}$ 时的归一化部分和，证明其依分布收敛，并刻画极限分布为临界高斯可乘混沌积分平方根与独立标准复正态变量的乘积。

A note on character sums over short moving intervals

We investigate the sums $(1/\sqrt{H}) \sum_{X < n \leq X+H} χ(n)$, where $χ$ is a fixed non-principal Dirichlet character modulo a prime $q$, and $0 \leq X \leq q-1$ is uniformly random. Davenport and Erdős, and more recently Lamzouri, proved central limit theorems for these sums provided $H \rightarrow \infty$ and $(\log H)/\log q \rightarrow 0$ as $q \rightarrow \infty$, and Lamzouri conjectured these should hold subject to the much weaker upper bound $H=o(q/\log q)$. We prove this is false for some $χ$, even when $H = q/\log^{A}q$ for any fixed $A > 0$. On the other hand, we show it is true…