Limit theorems on locally compact Abelian groups

We prove limit theorems for row sums of a rowwise independent infinitesimal array of random variables with values in a locally compact Abelian group. First we give a proof of Gaiser's theorem [4, Satz 1.3.6], since it does not have an easy access and it is not complete. This theorem gives sufficient conditions for convergence of the row sums, but the limit measure cannot have a nondegenerate idempotent factor. Then we prove necessary and sufficient conditions for convergence of the row sums, where the limit measure can be also a nondegenerate Haar measure on a compact subgroup. Finally, we investigate special cases: the torus group, the group of p ‐adic integers and the p ‐adic solenoid. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new look at Bergström’s theorem on convergence in distribution for sums of dependent random variables

In his 1972Periodica Mathematica Hungarica paper, H. Bergström stated a theorem on convergence in distribution for triangular arrays of dependent random variables satisfying, a ?-mixing condition. A gap in his proof of this theorem is explained and a more general version is proved under weakened hypotheses. The method used consists of comparisons between the given array and associated arrays which are parameterized by a truncation variable. In addition to the main theorem, this method yields a proof of equality of limiting finite-dimensional distributions for processes generated by the given associated arrays as well as the result that if a limit distribution for the centered row sums does exist, it must be infinitely divisible. Several corollaries to the main theorem specialize this result for convergence to distributions within certain subclasses of the infinitely divisible laws.     

Biproportional scaling of matrices and the iterative proportional fitting procedure

A short proof is given of the necessary and sufficient conditions for the convergence of the Iterative Proportional Fitting procedure. The input consists of a nonnegative matrix and of positive target marginals for row sums and for column sums. The output is a sequence of scaled matrices to approximate the biproportional fit, that is, the scaling of the input matrix by means of row and column divisors in order to fit row and column sums to target marginals. Generally it is shown that certain structural properties of a biproportional scaling do not depend on the particular sequence used to approximate it. Specifically, the sequence that emerges from the Iterative Proportional Fitting procedure is analyzed by means of the L ₁-error that measures how current row and column sums compare to their target marginals. As a new result a formula for the limiting L ₁-error is obtained. The formula is in terms of partial sums of the target marginals, and easily yields the other well-known convergence characterizations.     

Limit points of the iterative scaling procedure

The iterative scaling procedure (ISP) is an algorithm which computes a sequence of matrices, starting from some given matrix. The objective is to find a matrix ’proportional’ to the given matrix, having given row and column sums. In many cases, for example if the initial matrix is strictly positive, the sequence is convergent. It is known that the sequence has at most two limit points. When these are distinct, convergence to these two points can be slow. We give an efficient algorithm which finds the limit points, invoking the ISP only on subproblems for which the procedure is convergent.     

On some methods for entropy maximization and matrix scaling

We describe and survey in this paper iterative algorithms for solving the discrete maximum entropy problem with linear equality constraints. This problem has applications e.g. in image reconstruction from projections, transportation planning, and matrix scaling. In particular we study local convergence and asymptotic rate of convergence as a function of the iteration parameter. For the trip distribution problem in transportation planning and the equivalent problem of scaling a positive matrix to achieve a priori given row and column sums, it is shown how the iteration parameters can be chosen in an optimal way. We also consider the related problem of finding a matrix X, diagonally similar to a given matrix, such that corresponding row and column norms in X are all equal. Reports of some numerical tests are given.     

Random dense bipartite graphs and directed graphs with specified degrees

Let s and t be vectors of positive integers with the same sum. We study the uniform distribution on the space of simple bipartite graphs with degree sequence s in one part and t in the other; equivalently, binary matrices with row sums s and column sums t . In particular, we find precise formulae for the probabilities that a given bipartite graph is edge‐disjoint from, a subgraph of, or an induced subgraph of a random graph in the class. We also give similar formulae for the uniform distribution on the set of simple directed graphs with out‐degrees s and in‐degrees t . In each case, the graphs or digraphs are required to be sufficiently dense, with the degrees varying within certain limits, and the subgraphs are required to be sufficiently sparse. Previous results were restricted to spaces of sparse graphs. Our theorems are based on an enumeration of bipartite graphs avoiding a given set of edges, proved by multidimensional complex integration. As a sample application, we determine the expected permanent of a random binary matrix with row sums s and column sums t . © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Mixing Limit Theorems of Maximum of Absolute Sums and their Convergence Rates

Conditioned, in the sense of Rényi (Acta Math. Acad. Sci. Hungar. 9, 215–228 1958), limit theorem in the L_p-norm of the maximum of absolute sums of independent identically distributed random variables is established and its exact rate of convergence is given. The results are equivalent to establishing analogous results for the supremum of random functions of partial sums defined on C[0,1], i.e., the invariance principle. New methodologies are used to prove the results that are profoundly different from those used in Rényi (Acta Math. Acad. Sci. Hungar. 9, 215–228, 1958) and subsequent authors in proving the conditioned central limit theorem for partial sums.     

Convergence of approximate martingale arrays to mixtures of infinitely divisible distributions in locally compact abelian groups

Sufficient conditions are given for the stable weak convergence of the row sums of an approximate martingale triangular array to a mixture of infinitely divisible distributions on a locally compact abelian group.     

Series representation for semistable laws and their domains of semistable attraction

If the centered and normalized partial sums of an i.i.d. sequence of random variables converge in distribution to a nondegenerate limit then we say that this sequence belongs to the domain of attraction of the necessarily stable limit. If we consider only the partial sums which terminate atk _n wherek _n+1 ck _n then the sequence belongs to the domain of semistable attraction of the necessarily semistable limit. In this paper, we consider the case where the limiting distribution is nonnormal. We obtain a series representation for the partial sums which converges almost surely. This representation is based on the order statistics, and utilizes the Poisson process. Almost sure convergence is a useful technical device, as we illustrate with a number of applications.This research was supported by a research scholarship from the Deutsche Forschungsgemeinschaft (DFG).     

行－列可交换随机变量组列的极限定理

利用逆鞅、截尾等方法，我们得出行－列可交换随机变量组列的大数定律，作为推论，我们得到具有有限均值的行－列可交换无限组列满足强大数定律的充要条件是该组列的对角线元素不相关．再充分利用对称性及可交换性，我们得到对称可交换随机变量和的极限定理，并由此导出对称行－列可交换随机变量组列的完全收敛定理     

On the deviation between the distributions of sums and maximum sums of IID random vectors

Summary For a sequence of independent and identically distributed random vectors, with finite moment of order less than or equal to the second, the rate at which the deviation between the distribution functions of the vectors of partial sums and maximums of partial sums is obtained both when the sample size is fixed and when it is random, satisfying certain regularity conditions. When the second moments exist the rate is of ordern ^−1/4 (in the fixed sample size case). Two applications are given, first, we compliment some recent work of Ahmad (1979,J. Multivariate Anal.,9, 214–222) on rates of convergence for the vector of maximum sums and second, we obtain rates of convergence of the concentration functions of maximum sums for both the fixed and random sample size cases.     

Sampling contingency tables

We give polynomial time algorithms for random sampling from a set of contingency tables, which is the set of m×n matrices with given row and column sums, provided the row and column sums are sufficiently large with respect to m, n. We use this to approximately count the number of such matrices. These problems are of interest in Statistics and Combinatorics. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 10 , 487–506, 1997     

Break minimization

A break in a {0, 1}-matrix is defined as a 0 with at least one 1 to its left and at least one 1 to its right in the same row. This paper is concerned with {0, 1}-matrices with given column sums and an upper limit for the row sums. In addition, there are limits on the distance from the first to the last 1 in a row. The problem that is considered is to find a {0, 1}-matrix satisfying the conditions such that the total number of breaks is minimum. An algorithm for solving this problem is presented. Computational results illustrate the effectiveness of the algorithm. The investigation originated in a problem of crew rostering.     

Convolution operators on Lebesgue spaces of an n-dimensional cube

This paper is concerned with the problem of diagonally scaling a given nonnegative matrix a to one with prescribed row and column sums. The approach is to represent one of the two scaling matrices as the solution of a variational problem. This leads in a natural way to necessary and sufficient conditions on the zero pattern of a so that such a scaling exists. In addition the convergence of the successive prescribed row and column sum normalizations is established. Certain invariants under diagonal scaling are used to actually compute the desired scaled matrix, and examples are provided. Finally, at the end of the paper, a discussion of infinite systems is presented.     

A class of martingales with non-symmetric limit distributions

Summary A class of non-pathological martingales is exhibited to which current martingale central limit theory does not apply. The problem consists in the convergence of the normalized sums of squares being only in distribution. The class is that of degenerate U-statistics.This work was done while the first two authors were on Sabbatical leave at the Technion.Work supported in part by National Research Council of Canada, Grant No. A-9068Work supported in part by National Research Council of Canada, Grant No. A-9076     

Some bounds for Markov chains

In principle it is possible to characterize the long run behavior of any evolutionary game by finding an analytical expression for its limit probability distribution. However, it is cumbersome to do so when the state space is large and the rate of mutation is significant. This paper gives upper and lower bounds for the limit distribution, which are easy to compute. The bounds are expressed in terms of the maximal and minimal row sums of parts of the transition matrix.     

Approximate Predetermined Convergence Properties of the Gibbs Sampler

This article aims to provide a method for approximately predetermining convergence properties of the Gibbs sampler. This is to be done by first finding an approximate rate of convergence for a normal approximation of the target distribution. The rates of convergence for different implementation strategies of the Gibbs sampler are compared to find the best one. In general, the limiting convergence properties of the Gibbs sampler on a sequence of target distributions (approaching a limit) are not the same as the convergence properties of the Gibbs sampler on the limiting target distribution. Theoretical results are given in this article to justify that under conditions, the convergence properties of the Gibbs sampler can be approximated as well. A number of practical examples are given for illustration.     

Mixing normal approximations of vectors of sums and maximum sums

Summary The conditioned central limit theorem for the vector of maximum partial sums based on independent identically distributed random vectors is investigated and the rate of convergence is discussed. The conditioning is that of Rényi (1958,Acta Math. Acad. Sci. Hungar.,9, 215–228). Analogous results for the vector of partial sums are obtained. University of Petroleum and Minerals     

Poisson convergence in two dimensions with application to row and column exchangeable arrays

A Poisson convergence theorem is given for a sequence of simple point processes on the plane. The approach taken here is to formulate sufficient conditions for Poisson convergence in terms of the behaviour of two one-dimensional compensators. This limit theorem is then applied to obtain a functional Poisson convergence result for a sequence of row and column exchangeable arrays.     

Exact Laws for Sums of Order Statistics from a Generalized Pareto Distribution

Abstract

We select the kth order statistic from each row from a sequence of independent and identically distributed random variables from a distribution that generalizes the Pareto distribution. We then examine weighted sums of these order statistics to see whether or not Laws of Large Numbers with nonzero limits exist.