6,236 research outputs found
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 be a bounded domain and let %
%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 . Let be a probability measure on .
Denote by 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 on its domain. It is known that 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
, indexed by , 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 in general and
the spectral gap in particular. The operator is
the generator of a diffusion process with random jumps from the boundary, and
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
penalized by its occupation time at the origin, also known as a
homopolymer. For a fixed real parameter and time , 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
times the occupation time at the origin up to time . The case was
studied previously by Cranston and Molchanov arXiv:1508.06915. We consider the
case , which is intrinsically different only when the underlying walk
is recurrent, that is . Our main result is a scaling limit for the
distribution of the homopolymer on the time interval , as ,
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-
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
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