Arunava Mukherjea

Limit Theorems: Stochastic Matrices, Ergodic Markov Chains, and Measures on Semigroups*

Publisher Summary This chapter reviews problems on ergodicity of non-homogeneous and homogeneous Markov chains to apply several of the results on measures on semi-groups. The chapter considers limit theorems for probability measures on stochastic matrices by first showing how these problems come up naturally in studying random walks on simple geometric figures on the plane. The abstract theory is reviewed. Completely simple semi-groups are considered to study limit theorems for convolution products of probability measures, including generalizations of the well-known Paul Levy theorem on the equivalence of convergence in probability, convergence with probability one, and convergence in distribution for sums of independent random variables. An important class of semi-groups that have been widely studied by semi-groupists and probabilists is the class of completely simple semi-groups. A semi-group is called completely simple if it is simple and contains a primitive idempotent. Various theorems are proven in the chapter. Convergence problems for products of stochastic matrices naturally come up in various contexts in biology, economics, and various other applications. The chapter discusses the problem of tendency to consensus in an information exchanging operation.

Identification of the parameters of a trivariate normal vector by the distribution of the minimum

This paper continues the work started by Basu and Ghosh (J. Mult. Anal. (1978), 8, 413–429), by Gilliland and Hannan (J. Amer. Stat. Assoc. (1980), 75, No. 371, 651–654), and then continued on by Mukherjea and Stephens (Prob. Theory and Rel. Fields (1990), 84, 289–296), and Elnaggar and Mukherjea (J. Stat. Planning and Inference (1990), 78, 23–37). Let (X1, X2,..., Xn) be a multivariate normal vector with zero means, a common correlation ρ and variances σ21, σ22,..., σ2n such that the parameters ρ, σ21, σ22,..., s2n are unknown, but the distribution of the max{Xi: 1≤i≤n} (or equivalently, the distribution of the min{Xi: 1≤i≤n}) is known. The problem is whether the parameters are identifiable and then how to determine the (unknown) parameters in terms of the distribution of the maximum (or its density). Here, we solve this problem for general n. Earlier, this problem was considered only for n≤3. Identifiability problems in related contexts were considered earlier by numerous authors including: T. W. Anderson and S. G. Ghurye, A. A. Tsiatis, H. A. David, S. M. Berman, A. Nadas, and many others. We also consider here the case where the Xis have a common covariance instead of a common correlation.

TIGHTNESS OF PRODUCTS OF I.I.D. RANDOM MATRICES. II

SummaryIn this paper we present a necessary and sufficient condition for tightness of products of i.i.d. finite dimensional random nonnegative matrices. We give an example illustrating the use of our theorem and treat completely the case of 2×2 matrices. We also describe stationary solutions of the linear equationyn=Xnyn−1, n>0, in (Rd)+, whereX1,X2,... are i.i.d.d×d nonnegative matrices.

Identification of parameters by the distribution of the maximum random variable : the general multivariate normal case

SummaryLetX1,X2, ...,Xr ber independentn-dimensional random vectors each with a non-singular normal distribution with zero means and positive partial correlations. Suppose thatXi=(Xi1, ...,Xin) and the random vectorY=(Y1, ...,Yn), their maximum, is defined byYj=max{Xij:1≦i≦r}. LetW be another randomn-vector which is the maximum of another such family of independentn-vectorsZ1,Z2, ...,Zs. It is then shown in this paper that the distributions of theZis are simply a rearrangement of those of theZjs (and of course,r=s), whenever their maximaY andW have the same distribution. This problem was initially studied by Anderson and Ghurye [2] in the univariate and bivariate cases and motivated by a supply-demand problem in econometrics.

Recurrent random walks in nonnegative matrices: attractors of certain iterated function systems

SummaryIn this paper the structure of the set of recurrent points for random walks in finite dimensional nonnegative matrices is determined. The structural results are then used in understanding attractors of certain (not necessarily contractive) iterated function systems.

The problem of identification of parameters by the distribution of the maximum random variable: solution for the trivariate normal case

Let (Xi, Yi, Zi), i = 1, 2, ..., m, be a number of independent random vectors each with a non-singular trivariate normal distribution function with non-zero correlations and zero means. Let (X, Y, Z) be their maximum, i.e., X = maxiXi, Y = maxiYi, and Z = maxiZi. In this paper, we show that the distribution of (X, Y, Z) uniquely determines the parameters of the distributions of (Xi, Yi, Zi), 1

The problem of identification of parameters by the distribution of the maximum random variable

Suppose that X1, X2,..., Xn are independently distributed according to certain distributions. Does the distribution of the maximum of {X1, X2,..., Xn} uniquely determine their distributions? In the univariate case, a general theorem covering the case of Cauchy random variables is given here. Also given is an affirmative answer to the above question for general bivariate normal random variables with non-zero correlations. Bivariate normal random variables with nonnegative correlations were considered earlier in this context by T. W. Anderson and S. G. Ghurye.

On the Continuous Singularity of the Limit Distribution of Products of I.I.D. d×d Stochastic Matrices

This article gives sufficient conditions for the limit distribution of products of i.i.d. d×d random stochastic matrices, d finite and ≥2, to be continuous singular, when the support of the distribution of the individual random matrices is finite or countably infinite. Proofs are based on applications of the multivariate Central Limit Theorem.

Convolution products of nonidentical distributions on a topological semigroup

In this paper we present a number of sufficient conditions on a sequence of probability measuresµn on a locally compact (second countable) Hausdorff topological semigroupS that guarantee the weak convergence of the sequence of convolution productsµk,n ≡µk + 1 *···*µn (k

On the Distribution of the Limit of Products of I.I.D. 2×2 Random Stochastic Matrices

This article gives sufficient conditions for the limit distribution of products of i.i.d. 2×2 random stochastic matrices to be continuous singular, when the support of the distribution of the individual random matrices is finite.