An Empirical Process Interpretation of a Model of Species Survival

We study a model of species survival recently proposed by Michael and Volkov. We interpret it as a variant of empirical processes, in which the sample size is random and when decreasing, samples of smallest numerical values are removed. Micheal and Volkov proved that the empirical distributions converge to the sample distribution conditioned not to be below a certain threshold. We prove a functional central limit theorem for the fluctuations. There exists a threshold above which the limit process is Gaussian with variance bounded below by a positive constant, while at the threshold it is half-Gaussian.Comment: to appear in Stochastic Processes and their Application

Spectral Analysis of a Family of Second-Order Elliptic Operators with Nonlocal Boundary Condition Indexed by a Probabilty Measure

Let $D\subset R^d$ be a bounded domain and let $$L=\frac12\nabla\cdot a\nabla +b\cdot\nabla$$ % %L=\frac12\sum_{i,j=1}^da_{i,j}\frac{\partial^2}{\partial x_i\partial x_j}+\sum_{i=1}^db_i\frac{\partial}{\partial x_i}, % be a second order elliptic operator on $D$. Let $\nu$ be a probability measure on $D$. Denote by ${\mathcal L}$ the differential operator whose domain is specified by the following non-local boundary condition: {\mathcal D_{{\mathcal L}}}=\{f\in C^2(\ol{D}): \int_D f d\nu = f|_{\partial D}\}, and which coincides with $L$ on its domain. It is known that $\mathcal L$ possesses an infinite sequence of eigenvalues, and that with the exception of the zero eigenvalue, all eigenvalues have negative real part. Define the spectral gap of $\mathcal {L}$, indexed by $\nu$, by \gamma_1(\nu)\equiv\sup\{\re \lambda:0\neq \lambda is an eigenvalue for {\mathcal L}\}. In this paper we investigate the eigenvalues of $\mathcal L$ in general and the spectral gap $\gamma_1(\nu)$ in particular. The operator $\mathcal L$ is the generator of a diffusion process with random jumps from the boundary, and $\gamma_1(\nu)$ measures the exponential rate of convergence of this process to its invariant measure.Comment: To appear in the Journal of Functional Analysi

The subcritical phase for a homopolymer model

We study a model of continuous-time nearest-neighbor random walk on $\mathbb{Z}^d$ penalized by its occupation time at the origin, also known as a homopolymer. For a fixed real parameter $\beta$ and time $t>0$, we consider the probability measure on paths of the random walk starting from the origin whose Radon-Nikodym derivative is proportional to the exponent of the product $\beta$ times the occupation time at the origin up to time $t$. The case $\beta>0$ was studied previously by Cranston and Molchanov arXiv:1508.06915. We consider the case $\beta<0$, which is intrinsically different only when the underlying walk is recurrent, that is $d=1,2$. Our main result is a scaling limit for the distribution of the homopolymer on the time interval $[0,t]$, as $t\to\infty$, a result that coincides with the scaling limit for penalized Brownian motion due to Roynette and Yor. In two dimensions, the penalizing effect is asymptotically diminished, and the homopolymer scales to standard Brownian motion. Our approach is based on potential analytic and martingale approximation for the model. We also apply our main result to recover a scaling limit for a wetting model. We study the model through analysis of resolvents.Comment: 32 page

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-$b$ 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.Comment: Version 1.0, 25 pages. Keywords: Zeckendorf decompositions, positive linear recurrence relations, distribution of gaps, longest gap, Markov processe

Spectral response of an elastic sphere to dipolar point-sources

A stratified elastic sphere is excited by an harmonic dipolar source of arbitrary orientation and depth. The total field is expanded in series of vector spherical harmonics and then condensed into a convenient form of a displacement dyadic. The Haskell-Gilbert matrix method is employed to obtain the radial factor of the displacements for a multilayered sphere. The dependence of the field on the azimuth angle and the fault elements is obtained for the case of a double-couple at depth. Expressions are also developed for the radiation pattern of surface waves over a spherical stratified earth