Tightness of products of i.i.d. random matrices

Summary In 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 equationy _n=X_ny_n–1, n>0, in (R ^d )⁺, whereX ₁,X ₂,... are i.i.d.d×d nonnegative matrices.     

Probability measures on 2 × 2 stochastic matrices: some open problems

Let u be a probability measure on 2 × 2 stochastic matrices with finite support such that the sequence μⁿ, the nth convolution power of μ, weakly converges to a probability measure λ whose support consists of 2 × 2 stochastic matrices with identical rows. The probability measure λ can, therefore, be regarded as a measure on the unit interval [0,1]. In this paper, we discuss some open problems regarding when λ is continuous singular or absolutely continuous with respect to the Lebesgue measure on [0,1], and when λ determines μ uniquely.     

Continuous Singularity of the Weak Limit of Convolution Powers of a Discrete Probability Measure on 2 × 2 Stochastic Matrices

Let be a probability measure on n 2 × 2 stochastic matrices, n an arbitrary positive integer, and = (w) lim _n ⁿ , such that the support of consists of 2 × 2 stochastic matrices of rank one, and as such, can be regarded as a probability measure on [0, 1]. We present simple sufficient conditions for to be continuous singular w.r.t. the Lebesgue measure on [0, 1]. We also determine , given .     

Maximal Homomorphic Group Image and Convergence of Convolution Sequences on a Semigroup

Let be a probability measure generating a locally compact semigroup S. If the convolution sequence ⁿ is tight, in particular if S is compact, S admits a closed minimal ideal K. The convergence of ⁿ is characterized in terms of convergence of a homomorphic image (~) ⁿ on a factor group of the compact group G in the Rees–Suschkewitsch decomposition of K.     

Bioleaching of low-grade copper ores usingThiobacillus ferrooxidans

Applied Biochemistry and Biotechnology -     

Convergence in Distribution of Products of I.I.D. Nonnegative Matrices

Let (X _i) be a sequence of m × m i.i.d. stochastic matrices with distribution . Then ⁿ is the distribution of X _n X _n–1 ...X ₁. Simple sufficient conditions for the weak convergence of ( ⁿ ) are presented here. An extremely simple (and verifiable) necessary and sufficient condition is provided for m= 3. The method for m= 3 works for m> 3 even though calculations are more involved for higher values of m. We also discuss the purity of the limit distribution for m2.     

Invariant points of maps between Grassmannians

We show that for a very large class of integers and any map between Grassmannians, there is some -plane of which is mapped into a subspace of itself.