A new class of distributional transformations is introduced, characterized by equations relating function weighted expectations of test functions on a given distribution to expectations of the transformed distribution on the test function's higher order derivatives. The class includes the size and zero bias transformations, and when specializing to weighting by polynomial functions, relates distributional families closed under independent addition, and in particular the infinitely divisible distributions, to the family of transformations induced by their associated orthogonal polynomial systems. For these families, generalizing a well known property of size biasing, sums of independent variables are transformed by replacing summands chosen according to a multivariate distribution on its index set by independent variables whose distributions are transformed by members of that same family. A variety of the transformations associated with the classical orthogonal polynomial systems have as fixed points the original distribution, or a member of the same family with different parameter.
In this paper, we extend Stein's method to products of independent beta, gamma, generalised gamma and mean zero normal random variables. In particular, we obtain Stein operators for mixed products of these distributions, which include the classical beta, gamma and normal Stein operators as special cases. These operators lead us to closed-form expressions involving the Meijer $G$-function for the probability density function and characteristic function of the mixed product of independent beta, gamma and central normal random variables.
This survey article discusses the main concepts and techniques of Stein's method for distributional approximation by the normal, Poisson, exponential, and geometric distributions, and also its relation to concentration inequalities. The material is presented at a level accessible to beginning graduate students studying probability with the main emphasis on the themes that are common to these topics and also to much of the Stein's method literature.
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.
For a family of linear preferential attachment graphs, we provide rates of convergence for the total variation distance between the degree of a randomly chosen vertex and an appropriate power law distribution as the number of vertices tends to infinity. Our proof uses a new formulation of Stein's method for the negative binomial distribution, which stems from a distributional transformation that has the negative binomial distributions as the only fixed points.
In this paper we extend Stein's method to the distribution of the product of $n$ independent mean zero normal random variables. A Stein equation is obtained for this class of distributions, which reduces to the classical normal Stein equation in the case $n=1$. This Stein equation motivates a generalisation of the zero bias transformation. We establish properties of this new transformation, and illustrate how they may be used together with the Stein equation to assess distributional distances for statistics that are asymptotically distributed as the product of independent central normal random…
Edgeworth expansion provides higher-order corrections to the normal approximation for a probability distribution. The classical proof of Edgeworth expansion is via characteristic functions. As a powerful method for distributional approximations, Stein's method has also been used to prove Edgeworth expansion results. However, these results assume that either the test function is smooth (which excludes indicator functions of the half line) or that the random variables are continuous (which excludes random variables having only a continuous component). Thus, how to recover the classical Edgeworth…
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…
Distributional transformations characterized by equations relating expectations of test functions weighted by a given biasing function on the original distribution to expectations of the test function's higher derivatives with respect to the transformed distribution play a great role in Stein's method and were, in great generality, first considered by Goldstein and Reinert \cite{GolRei05b}. We prove two abstract existence and uniqueness results for such distributional transformations, generalizing their $X-P$ bias transformation. On the one hand, we show how one can abandon previously necessar…
Combining Stein's method with heat kernel techniques, we study the function Tr(AO), where A is a fixed n by n real matrix over such that Tr(AA^t)=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 2 * squareroot(2) /(n-1), slightly improving the constant in a bound of Meckes, which was obtained by completely different methods.
The variance-gamma (VG) distributions form a four parameter family that includes as special and limiting cases the normal, gamma and Laplace distributions. Some of the numerous applications include financial modelling and approximation on Wiener space. Recently, Stein's method has been extended to the VG distribution. However, technical difficulties have meant that bounds for distributional approximations have only been given for smooth test functions (typically requiring at least two derivatives for the test function). In this paper, which deals with symmetric variance-gamma (SVG) distributio…
We build upon recent advances on the distributional aspect of Stein's method to propose a novel and flexible technique for computing Stein operators for random variables that can be written as products of independent random variables. We show that our results are valid for a wide class of distributions including normal, beta, variance-gamma, generalized gamma and many more. Our operators are $k$th degree differential operators with polynomial coefficients; they are straightforward to obtain even when the target density bears no explicit handle. As an application, we derive a new formula for th…
We develop Stein's method for the half-normal distribution and apply it to derive rates of convergence in distributional limit theorems for three statistics of the simple symmetric random walk: the maximum value, the number of returns to the origin and the number of sign changes up to a given time $n$. We obtain explicit error bounds with the optimal rate $n^{-1/2}$ for both the Kolmogorov and the Wasserstein metric. In order to apply Stein's method, we compare the characterizing operator of the limiting half-normal distribution with suitable characterizations of the discrete approximating dis…
We introduce two abstract theorems that reduce a variety of complex exponential distributional approximation problems to the construction of couplings. These are applied to obtain new rates of convergence with respect to the Wasserstein and Kolmogorov metrics for the theorem of Rényi on random sums and generalizations of it, hitting times for Markov chains, and to obtain a new rate for the classical theorem of Yaglom on the exponential asymptotic behavior of a critical Galton--Watson process conditioned on nonextinction. The primary tools are an adaptation of Stein's method, Stein couplings, a…
We propose a new general version of Stein's method for univariate distributions. In particular we propose a canonical definition of the Stein operator of a probability distribution {which is based on a linear difference or differential-type operator}. The resulting Stein identity highlights the unifying theme behind the literature on Stein's method (both for continuous and discrete distributions). Viewing the Stein operator as an operator acting on pairs of functions, we provide an extensive toolkit for distributional comparisons. Several abstract approximation theorems are provided. Our appro…
We obtain rates of convergence in limit theorems of partial sums $S_n$ for certain sequences of dependent, identically distributed random variables, which arise naturally in statistical mechanics, in particular, in the context of the Curie-Weiss models. Under appropriate assumptions there exists a real number $α$, a positive real number $μ$, and a positive integer $k$ such that $(S_n- n α)/n^{1 - 1/2k}$ converges weakly to a random variable with density proportional to $\exp(-μ|x|^{2k} /(2k)!)$. We develop Stein's method for exchangeable pairs for a rich class of distributional approximations …
With ${\mathcal{Q}}_{q,n}$ the distribution of $n$ minus the rank of a matrix chosen uniformly from the collection of all $n\times(n+m)$ matrices over the finite field $\mathbb{F}_q$ of size $q\ge2$, and ${\mathcal{Q}}_q$ the distributional limit of ${\mathcal{Q}}_{q,n}$ as $n\rightarrow\infty$, we apply Stein's method to prove the total variation bound $\frac{1}{8q^{n+m+1}}\leq\|{\mathcal{Q}}_{q,n}-{\mathcal{Q}}_q\|_{\mathrm{TV}}\leq\frac{3}{q^{n+m+1}}$. In addition, we obtain similar sharp results for the rank distributions of symmetric, symmetric with zero diagonal, skew symmetric, skew cen…
Stein's method has been widely used to achieve distributional approximations for probability distributions defined in Euclidean spaces. Recently, techniques to extend Stein's method to manifold-valued random variables with distributions defined on the respective manifolds have been reported. However, several of these methods impose strong regularity conditions on the distributions as well as the manifolds and/or consider very special cases. In this paper, we present a novel framework for Stein's method on Riemannian manifolds using the Friedrichs extension technique applied to self-adjoint unb…
The variance-gamma (VG) distributions form a four-parameter family which includes as special and limiting cases the normal, gamma and Laplace distributions. Some of the numerous applications include financial modelling and distributional approximation on Wiener space. In this review, we provide an up-to-date account of the basic distributional theory of the VG distribution. Properties covered include probability and cumulative distribution functions, generating functions, moments and cumulants, mode and median, Stein characterisations, representations in terms of other random variables, and a …
By the continuous mapping theorem, if a sequence of $d$-dimensional random vectors $(\mathbf{W}_n)_{n\geq1}$ converges in distribution to a multivariate normal random variable $Σ^{1/2}\mathbf{Z}$, then the sequence of random variables $(g(\mathbf{W}_n))_{n\geq1}$ converges in distribution to $g(Σ^{1/2}\mathbf{Z})$ if $g:\mathbb{R}^d\rightarrow\mathbb{R}$ is continuous. In this paper, we develop Stein's method for the problem of deriving explicit bounds on the distance between $g(\mathbf{W}_n)$ and $g(Σ^{1/2}\mathbf{Z})$ with respect to smooth probability metrics. We obtain several bounds for t…
We study a new class of time inhomogeneous Pólya-type urn schemes and give optimal rates of convergence for the distribution of the properly scaled number of balls of a given color to nearly the full class of generalized gamma distributions with integer parameters, a class which includes the Rayleigh, half-normal and gamma distributions. Our main tool is Stein's method combined with characterizing the generalized gamma limiting distributions as fixed points of distributional transformations related to the equilibrium distributional transformation from renewal theory. We identify special cases …
In a random graph, counts for the number of vertices with given degrees will typically be dependent. We show via a multivariate normal and a Poisson process approximation that, for graphs which have independent edges, with a possibly inhomogeneous distribution, only when the degrees are large can we reasonably approximate the joint counts as independent. The proofs are based on Stein's method and the Stein-Chen method with a new size-biased coupling for such inhomogeneous random graphs, and hence bounds on distributional distance are obtained. Finally we illustrate that apparent (pseudo-) powe…
We consider the cost of general orthogonal range queries in random quadtrees. The cost of a given query is encoded into a (random) function of four variables which characterize the coordinates of two opposite corners of the query rectangle. We prove that, when suitably shifted and rescaled, the random cost function converges uniformly in probability towards a random field that is characterized as the unique solution to a distributional fixed-point equation. We also state similar results for $2$-d trees. Our results imply for instance that the worst case query satisfies the same asymptotic esti…
A distributional symmetry is invariance of a distribution under a group of transformations. Exchangeability and stationarity are examples. We explain that a result of ergodic theory provides a law of large numbers: If the group satisfies suitable conditions, expectations can be estimated by averaging over subsets of transformations, and these estimators are strongly consistent. We show that, if a mixing condition holds, the averages also satisfy a central limit theorem, a Berry-Esseen bound, and concentration. These are extended further to apply to triangular arrays, to randomly subsampled ave…
