Stein's Method Research Articles

On the Almost Sure Central Limit Theorem for a Class of Z d -Actions

As an extension of earlier papers on stationary sequences, a concept of weak dependence for strictly stationary random fields is introduced in terms of so-called homoclinic transformations. Under assumptions made within the framework of this concept a form of the almost sure central limit theorem (ASCLT) is established for random fields arising from a class of algebraic Zd-actions on compact abelian groups. As an auxillary result, the central limit theorem is proved via Ch. Stein's method. The next stage of the proof includes some estimates which are specific for ASCLT. Both steps are based on making use of homoclinic transformations.

Estimate of the Accuracy of the Compound Poisson Approximation for the Distribution of the Number of Matching Patterns

Let $X_1,\ldots,X_m$ and $Y_1,\ldots,Y_n$ be two sequences of independent identically distributed random variables taking on values $1, 2,\ldots\;$. By means of a particular version of the Stein method we construct an estimate of the accuracy of approximation for the distribution of the number of matching patterns of outcomes $X_i,\ldots,X_{i+s-1}$ of a given length~s in the first sequence with the patterns of outcomes $Y_j,\ldots,Y_{j+s-1}$ in the second sequence. The approximating distribution is the distribution of the sum of Poisson number of independent random variables with geometric distribution.

The number of two-dimensional maxima

Let n points be placed uniformly at random in a subset A of the plane. A point is said to be maximal in the configuration if no other point is larger in both coordinates. We show that, for large n and for many sets A, the number of maximal points is approximately normally distributed. The argument uses Stein's method, and is also applicable in higher dimensions.

The number of two-dimensional maxima

Letnpoints be placed uniformly at random in a subsetAof the plane. A point is said to be maximal in the configuration if no other point is larger in both coordinates. We show that, for largenand for many setsA, the number of maximal points is approximately normally distributed. The argument uses Stein's method, and is also applicable in higher dimensions.

Normal approximation for quasi-associated random fields

For quasi-associated random fields (comprising negatively and positively dependent fields) on Z d we use Stein's method to establish the rate of normal approximation for partial sums taken over arbitrary finite subsets U of Z d .

Stein's Method and Birth-Death Processes

Barbour introduced a probabilistic view of Stein's method for estimating the error in probability approximations. However, in the case of approximations by general distributions on the integers, there have been no purely probabilistic proofs of Stein's bounds till this paper. Furthermore, the methods introduced here apply to a very large class of approximating distributions on the non-negative integers, among which there is a natural class for higher-order approximations by probability distributions rather than signed measures (as previously). The methods also produce Stein magic factors for process approximations which do not increase with the window of observation and which are simpler to apply than those in Brown, Weinberg and Xia.

A non-uniform Berry–Esseen bound via Stein's method

This paper is part of our efforts to develop Stein's method beyond uniform bounds in normal approximation. Our main result is a proof for a non-uniform Berry–Esseen bound for independent and not necessarily identically distributed random variables without assuming the existence of third moments. It is proved by combining truncation with Stein's method and by taking the concentration inequality approach, improved and adapted for non-uniform bounds. To illustrate the technique, we give a proof for a uniform Berry–Esseen bound without assuming the existence of third moments.

Orthogonal Polynomials in Stein's Method

Stein's method provides a way of finding approximations to the distribution, ρ say, of a random variable, which at the same time gives estimates of the approximation error involved. In essence the method is based on a defining equation, or equivalently an operator, of the distribution ρ and a related Stein equation. Up to now it was not clear which equation to take. One could think of a lot of equations. We show how for a broad class of distributions, there is one convenient equation. We give a systematic treatment, including the Stein equation and its solution. A key tool in Stein's theory is the generator method developed by Barbour and we suggest employing the generator of a Markov process for the operator of the Stein equation. For a given distribution there may be various Markov processes which fit in Barbour's method and, up to now, it was still not clear which Markov process to take to obtain good results. We show how for a broad class of distributions there is a special Markov process, a birth and death process, or a diffusion, which takes a leading role in the analysis. Furthermore, a key role is played by the classical orthogonal polynomials. It turns out that the defining operator is based on a hypergeometric difference or differential equation, which lies at the heart of the classical orthogonal polynomials. Furthermore, the spectral representation of the transition probabilities of the Markov process involved are in terms of orthogonal polynomials closely related to the distribution to be approximated. This systematic treatment together with the introduction of orthogonal polynomials in the analysis seems to be new. Furthermore some earlier uncovered examples like the beta, the Student's t, and the hypergeometric distribution are now worked out.

Motoo's combinatorial central limit theorem for serial rank statistics

Abstract We shall use the Stein method to prove a general Motoo-type combinatorial central limit theorem for serial rank statistics. Our basic approach will be the underlying graph structure of the ranks of lag 1. In the process we shall obtain minimal conditions for the asymptotic normality of Wald–Wolfowitz-type serial rank statistics.

On Compound Poisson Approximation for Sequence Matching

Consider sequences {Xi}mi=1 and {Yj}nj=1 of independent random variables, taking values in a finite alphabet, and assume that the variables X1, X2, … and Y1, Y2, … follow the distributions μ and v, respectively. Two variables Xi and Yj are said to match if Xi = Yj. Let the number of matching subsequences of length k between the two sequences, when r, 0 [les ] r < k, mismatches are allowed, be denoted by W.In this paper we use Stein's method to bound the total variation distance between the distribution of W and a suitably chosen compound Poisson distribution. To derive rates of convergence, the case where E[W] stays bounded away from infinity, and the case where E[W] → ∞ as m, n → ∞, have to be treated separately. Under the assumption that ln n/ln(mn) → ρ ∈ (0, 1), we give conditions on the rate at which k → ∞, and on the distributions μ and v, for which the variation distance tends to zero.

Estimating Stein's Constants for Compound Poisson Approximation

Stein's method for compound Poisson approximation was introduced by Barbour, Chen and Loh. One difficulty in applying the method is that the bounds on the solutions of the Stein equation are by no means as good as for Poisson approximation. We show that, for the Kolmogorov metric and under a condition on the parameters of the approximating compound Poisson distribution, bounds comparable with those obtained for the Poisson distribution can be recovered.

On stationary renewal reward processes where most rewards are zero

We consider a stationary version of a renewal reward process, i.e., a renewal process where a random variable called a reward is associated with each renewal. The rewards are nonnegative and I.I.D., but each reward may depend on the distance to the next renewal. We give an explicit bound for the total variation distance between the distribution of the accumulated reward over the interval (0,L] and a compound Poisson distribution. The bound depends in its simplest form only on the first two joint moments of T and Y (or I{Y > 0}), where T is the distance between successive renewals and Y is the reward. If T and Y are independent, and LE(Y) (or LP(Y > 0)) is bounded or Y binary valued, then the bound is O(E(Y)) as E(Y) → 0 (or O(P(Y > 0)) as P(Y > 0) → 0). To prove our result we generalize a Poisson approximation theorem for point processes by Barbour and Brown, derived using Stein's method and Palm theory, to the case of compound Poisson approximation, and combine this theorem with suitable couplings.

On the central limit theorem for negatively correlated random variables with negatively correlated squares

Using Stein's method, assuming Lindeberg's condition, we find a necessary and sufficient condition for the central limit theorem to hold for an array of random variables such that the variables in each row are negatively correlated (i.e., every pair has negative covariance) and their squares are also negatively correlated (in fact, a somewhat more general result is shown). In particular, we obtain a necessary and sufficient condition for the central limit theorem to hold for an array of pairwise independent random variables satisfying Lindeberg's condition. A collection of random variables is said to be jointly symmetric if finite-dimensional joint distributions do not change when a subset of the variables is multiplied by −1. A corollary of our main result is that the central limit theorem holds for pairwise independent jointly symmetric random variables under Lindeberg's condition. We also prove a central limit theorem for a triangular array of variables satisfying some size constraints and where the n variables in each row are φ( n)-tuplewise independent, i.e., every subset of cardinality no greater than φ( n) is independent, where φ is a function such that φ( n)/ n 1/2→∞.

Normal approximations by Stein's method

Stein's method for normal approximations is explained, with some examples and applications. In the study of the asymptotic distribution of the sum of dependent random variables, Stein's method may be a very useful tool. We have attempted to write an elementary introduction. For more advanced introductions to Stein's method, see Stein (1986), Barbour (1997) and Chen (1998).

Poisson approximation for large contours in low-temperature Ising models

We consider the contour representation of the infinite volume Ising model at any fixed inverse temperature β>β ∗ , the solution of ∑ θ: θ∋0 e −β|θ|=1 . Let μ be the infinite-volume “+” measure. Fix V⊂ Z d , λ>0 and a (large) N such that calling G N,V the set of contours of length at least N intersecting V, there are in average λ contours in G N,V under μ. We show that the total variation distance between the law of (γ : γ∈ G N,V) under μ and a Poisson process is bounded by a constant depending on β and λ times e −(β−β ∗)N . The proof builds on the Chen–Stein method as presented by Arratia, Goldstein and Gordon. The control of the correlations is obtained through the loss-network space-time representation of contours due to Fernández, Ferrari and Garcia.

Compound Poisson approximation and the clustering of random points

Let n random points be uniformly and independently distributed in the unit square, and count the number W of subsets of k of the points which are covered by some translate of a small square C. If n|C| is small, the number of such clusters is approximately Poisson distributed, but the quality of the approximation is poor. In this paper, we show that the distribution of W can be much more closely approximated by an appropriate compound Poisson distribution CP(λ1, λ2,…). The argument is based on Stein's method, and is far from routine, largely because the approximating distribution does not satisfy the simplifying condition that iλ i be decreasing.

Compound Poisson approximation and the clustering of random points

Let n random points be uniformly and independently distributed in the unit square, and count the number W of subsets of k of the points which are covered by some translate of a small square C. If n|C| is small, the number of such clusters is approximately Poisson distributed, but the quality of the approximation is poor. In this paper, we show that the distribution of W can be much more closely approximated by an appropriate compound Poisson distribution CP(λ1, λ2,…). The argument is based on Stein's method, and is far from routine, largely because the approximating distribution does not satisfy the simplifying condition that iλi be decreasing.

Probabilistic and statistical properties of words: an overview.

In the following, an overview is given on statistical and probabilistic properties of words, as occurring in the analysis of biological sequences. Counts of occurrence, counts of clumps, and renewal counts are distinguished, and exact distributions as well as normal approximations, Poisson process approximations, and compound Poisson approximations are derived. Here, a sequence is modelled as a stationary ergodic Markov chain; a test for determining the appropriate order of the Markov chain is described. The convergence results take the error made by estimating the Markovian transition probabilities into account. The main tools involved are moment generating functions, martingales, Stein's method, and the Chen-Stein method. Similar results are given for occurrences of multiple patterns, and, as an example, the problem of unique recoverability of a sequence from SBH chip data is discussed. Special emphasis lies on disentangling the complicated dependence structure between word occurrences, due to self-overlap as well as due to overlap between words. The results can be used to derive approximate, and conservative, confidence intervals for tests.

Iterates of expanding maps

The iterates of expanding maps of the unit interval into itself have many of the properties of a more conventional stochastic process, when the expanding map satisfies some regularity conditions and when the starting point is suitably chosen at random. In this paper, we show that the sequence of iterates can be closely tied to an m-dependent process. This enables us to prove good bounds on the accuracy of Gaussian approximations. Our main tools are coupling and Stein's method.

Compound Poisson approximations of subgraph counts in random graphs

Poisson approximation, random graphs, Stein's method Poisson approximations for the counts of a given subgraph in large random graphs were accomplished using Stein's method by Barbour and others. Compound Poisson approximation results, on the other hand, have not appeared, at least partly because of the lack of a suitable coupling. We address that problem by introducing the concept of cluster determining pairs, leading to a useful coupling for a large class of subgraphs we call local. We find bounds on the compound Poisson approximation of counts of local subgraphs in large random graphs.