Convergence of nonhomogeneous stochastic chains with countable states

The work started by V. M. Maksimov [1970, Theory Probab. Appl.15, 604–618], and continued by A. Mukherjea [1980, Trans. Amer. Math. Soc.263, 505–520], is extended, and completed with respect to certain aspects. Infinite-dimensional stochastic chains are considered in the framework of Mukherjea [loc. cit.]; backward products of stochastic matrices and their convergence are also considered. The main theme centers around understanding how the convergence of products (backward and forward, finite and infinite dimensional) takes place and what it means in terms of various types of asymptotic behavior of the individual stochastic matrices in the chain. The study is based on establishing the existence of a basis for convergent chains. The basis then makes it possible to describe properly various aspects of convergence. All results are new; they are also complete at least in the sense they have been presented and suitable examples (or counter-examples) are presented to justify the assumptions involved.     

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

Summary LetX ₁,X ₂, ...,X _r ber independentn-dimensional random vectors each with a non-singular normal distribution with zero means and positive partial correlations. Suppose thatX _i =(X _i1 , ...,X _in ) and the random vectorY=(Y ₁, ...,Y _n ), their maximum, is defined byY _j =max{X _ij :1ir}. LetW be another randomn-vector which is the maximum of another such family of independentn-vectorsZ ₁,Z ₂, ...,Z _s . It is then shown in this paper that the distributions of theZ _i 's are simply a rearrangement of those of theZ _j 's (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.     

Solution of a problem on the identification of parameters by the distribution of the maximum random variable: a multivariate normal case

In this paper we solve the problem of unique factorization of products ofn-variate nonsingular normal distributions with covariance matrices of the form , _ij =p _{i j} forij, = _i ² ,j=j,p0.     

On certain conjectures on invariant measures on semigroups

A conjecture stating that a locally compact semigroup admits a twosided semi-invariant measure iff it contains a kernel which is a unimodular group, is proven. Also a conjecture stating that the support of an r^*-invariant measure is a left group, is proven under the condition that for some a ε F (=support of the measure), aF is right cancellative. Moreover four types of invariance for regular probability measures are shown to be equivalent. Also a new proof of the equivalence of a two-sided semi-invariant probability measure and the existence of a kernel which is a compact group, is given.     

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

Suppose that X₁, X₂,…, X_n are independently distributed according to certain distributions. Does the distribution of the maximum of {X₁, X₂,…, X_n} 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.     

Upper packing dimension of a measure and the limit distribution of products of i.i.d. stochastic matrices

This article gives sufficient conditions for the limit distribution of products of i.i.d. 2 × 2 stochastic matrices to be continuous singular, when the support of the distribution of the individual random matrices is countably infinite. It extends a previous result for which the support of the random matrices is finite. The result is based on adapting existing proofs in the context of attractors and iterated function systems to the case of infinite iterated function systems.