Iddo Ben-Ari

A PROBABILISTIC APPROACH TO GENERALIZED ZECKENDORF DECOMPOSITIONS

Generalized Zeckendorf decompositions are expansions of integers as sums of elements of solutions to recurrence relations. The simplest cases are base-

Probabilistic approach to Perron root, the group inverse, and applications

b

Maclaurin's Inequality and a Generalized Bernoulli Inequality

expansions, and the standard Zeckendorf decomposition uses the Fibonacci sequence. The expansions are finite sequences of nonnegative integer coefficients (satisfying certain technical conditions to guarantee uniqueness of the decomposition) and which can be viewed as analogs of sequences of variable-length words made from some fixed alphabet. In this paper we present a new approach and construction for uniform measures on expansions, identifying them as the distribution of a Markov chain conditioned not to hit a set. This gives a unified approach that allows us to easily recover results on the expansions from analogous results for Markov chains, and in this paper we focus on laws of large numbers, central limit theorems for sums of digits, and statements on gaps (zeros) in expansions. We expect the approach to prove useful in other similar contexts.

Stochastic analysis of the motion of DNA nanomechanical bipeds

A probabilistic approach to the study of the Perron root of irreducible nonnegative matrices is presented. Our two main results are a probabilistic representation for the generalized inverse of the generator of a continuous-time finite-state Markov chain and the identification of the Hessian of the Perron root of a nonnegative irreducible matrix, with respect to a certain natural transformation of its entries, as a covariance matrix for additive functionals for a related Markov chain. These provide us with a natural approach – at least from the point of view of the probabilist – to study positivity and convexity properties of perturbations to the Perron eigenvalue. We focus on reestablishing and improving a number of known results in the field.

On Wallis-type Products and Pólya's Urn Schemes

Summary Maclaurins inequality is a natural, but nontrivial, generalization of the arithmetic-geometric mean inequality. We present a new proof that is based on an analogous generalization of Bernoullis inequality. Applications of Maclaurins inequality to iterative sequences and probability are discussed, along with graph-theoretic versions of the Maclaurin and Bernoulli inequalities.

On a Local Version of the Bak–Sneppen Model

In this paper, we formulate and analyze a Markov process modeling the motion of DNA nanomechanical walking devices. We consider a molecular biped restricted to a well-defined one-dimensional track and study its asymptotic behavior. Our analysis allows for the biped legs to be of different molecular composition, and thus to contribute differently to the dynamics. Our main result is a functional central limit theorem for the biped with an explicit formula for the effective diffusivity coefficient in terms of the parameters of the model. A law of large numbers, a recurrence/transience characterization and large deviations estimates are also obtained. Our approach is applicable to a variety of other biological motors such as myosin and motor proteins on polymer filaments.

Ergodic behavior of diffusions with random jumps from the boundary

Abstract A famous “curious identity” of Wallis gives a representation of the constant π in terms of a simply structured infinite product of fractions. Sondow and Yi [Amer. Math. Monthly 117 (2010) 912–917] identified a general scheme for evaluating Wallis-type infinite products. The main purpose of this paper is to discuss an interpretation of the scheme by means of Pólya urn models.

Spectral analysis of a family of second-order elliptic operators with nonlocal boundary condition indexed by a probability measure

A major difficulty in studying the Bak–Sneppen model is in effectively comparing it with well-understood models. This stems from the use of two geometries: complete graph geometry to locate the global fitness minimizer, and graph geometry to replace the species in the neighborhood of the minimizer. Over the years a number of models inspired by Bak–Sneppen were studied, usually by introducing different or new features (e.g. discretizing fitness, randomized neighbors or population size). We present a variant that only uses features present in Bak–Sneppen, and whose difference from the Bak–Sneppen is that only the graph geometry is used for the evolution. This allows to obtain the stationary distribution through random walk dynamics while preserving the geometric nature of the model. We use this to show that for constant-degree graphs, the stationary fitness distribution converges to an IID law as the number of vertices tends to infinity. We also discuss exponential ergodicity through coupling, and avalanches for the model.