Eric S. Key

Lower bounds for the maximal Lyapunov exponent

Upper bounds for the maximal Lyapunov exponent,E, of a sequence of matrix-valued random variables are easy to come by asE is the infimum of a real-valued sequence. We shall show that under irreducibility conditions similar to those needed to prove the Perron-Frobenius theorem, one can find sequences which increase toE. As a byproduct of the proof we shall see that we may replace the matrix norm with the spectral radius when computingE in such cases. Finally, a sufficient condition for transience of random walk in a random environment is given.

Computable examples of the maximal Lyapunov exponent

SummarySome new examples are given of sequences of matrix valued random variables for which it is possible to compute the maximal Lyapunov exponent. The examples are constructed by using a sequence of stopping times to group the original sequence into commuting blocks. If the original sequence is the outcome of independent Bernoulli trials with success probability p, then the maximal Lyapunov exponent may be expressed in terms of power series in p, with explicit formulae for the coefficients. The convexity of the maximal Lyapunov exponent as a function of p is discussed, as is an application to branching processes in a random environment.

Eigenvalue multiplicities of products of companion matrices.

It is shown that under suitable conditions an eigenvalue of a product of companion matrices has geometric multiplicity equal to one. The result is used to show that for a class of Random Walks in Periodic Environments recurrence is equivalent to a product of companion matrices having 1 as an eigenvalue of algebraic multiplicity greater than one.

Compositions of Random Möbius Transformations

Abstract Let be a sequence of i.i.d. random variables taking values in the set of all Möbius transformations. Consider the sequence defined by where z belongs to the extended complex plane. Necessary and sufficient conditions for a.s. convergence of will be discussed. As an application we will see how a sequence of the form may be used to generalize a mathematical model of phyllotaxis. Some results concerning the convergence in distribution of will also be discussed.

Some characterizations of the normal distribution

Various characterizations of the univariate normal distribution are presented. c 1999 Elsevier Science B.V. All rights reserved

Divergence rates for the number of rare numbers

Suppose thatX1,X2, ... is a sequence of i.i.d. random variables taking value inZ+. Consider the random sequenceA(X)≡(X1,X2,...). LetYn be the number of integers which appear exactly once in the firstn terms ofA(X). We investigate the limit behavior ofYn/E[Yn] and establish conditions under which we have almost sure convergence to 1. We also find conditions under which we dtermine the rate of growth ofE[Yn]. These results extend earlier work by the author.

A Painless Approach to Least Squares

(σ x) 2 < nσ x 2. Therefore, D = f mm f bb − (f mb) 2 = 4(nσ x 2 − (σ x) 2) > 0, and we are done. Acknowledgment. I wish to thank the referee for helpful suggestions and comments.

A recipe for bivariate copulas

ABSTRACT We give conditions on a ⩾ −1, b ∈ ( − ∞, ∞), and f and g so that Ca, b(x, y) = xy(1 + af(x)g(y))b is a bivariate copula. Many well-known copulas are of this form, including the Ali–Mikhail–Haq Family, Huang–Kotz Family, Bairamov–Kotz Family, and Bekrizadeh–Parham–Zadkarmi Family. One result is that we produce an algorithm for producing such copulas. Another is a one-parameter family of copulas whose measures of concordance range from 0 to 1.

A note on the spectral radius of a product of companion matrices

Conditions are given on the coefficients of the characteristic polynomials of a set of k companion matrices to ensure that the spectral radius of their product is bounded by t k where 0 < t < 1.

Perturbation of orthonormal bases inL2-spaces

The paper improves and generalizes a classical result from Paley and Wiener in their book on Fourier transforms. Paley and Wiener gave conditions on functionshn that imply that the sequence (1+hn(x))einx is a Riesz basis forL2[−π,π]. These conditions involve theL2-norm of the second derivativeshn″. The new results replace the differential operatory→y″ by more general differential operators inL2-spaces, in particular, by the Hermite differential operator inL2(R), andeinx by arbitrary orthonormal bases.