54,991 research outputs found
Weak Disorder in Fibonacci Sequences
We study how weak disorder affects the growth of the Fibonacci series. We
introduce a family of stochastic sequences that grow by the normal Fibonacci
recursion with probability 1-epsilon, but follow a different recursion rule
with a small probability epsilon. We focus on the weak disorder limit and
obtain the Lyapunov exponent, that characterizes the typical growth of the
sequence elements, using perturbation theory. The limiting distribution for the
ratio of consecutive sequence elements is obtained as well. A number of
variations to the basic Fibonacci recursion including shift, doubling, and
copying are considered.Comment: 4 pages, 2 figure
Tightness for a family of recursion equations
In this paper we study the tightness of solutions for a family of recursion
equations. These equations arise naturally in the study of random walks on
tree-like structures. Examples include the maximal displacement of a branching
random walk in one dimension and the cover time of a symmetric simple random
walk on regular binary trees. Recursion equations associated with the
distribution functions of these quantities have been used to establish weak
laws of large numbers. Here, we use these recursion equations to establish the
tightness of the corresponding sequences of distribution functions after
appropriate centering. We phrase our results in a fairly general context, which
we hope will facilitate their application in other settings.Comment: Published in at http://dx.doi.org/10.1214/08-AOP414 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
A metapopulation model with Markovian landscape dynamics
We study a variant of Hanski's incidence function model that allows habitat
patch characteristics to vary over time following a Markov process. The widely
studied case where patches are classified as either suitable or unsuitable is
included as a special case. For large metapopulations, we determine a recursion
for the probability that a given habitat patch is occupied. This recursion
enables us to clarify the role of landscape dynamics in the survival of a
metapopulation. In particular, we show that landscape dynamics affects the
persistence and equilibrium level of the metapopulation primarily through its
effect on the distribution of a local population's life span.Comment: This manuscript version is made available under the CC-BY-NC-ND 4.0
license http://creativecommons.org/licenses/by-nc-nd/4.0
Computational Dynamic Market Risk Measures in Discrete Time Setting
Different approaches to defining dynamic market risk measures are available
in the literature. Most are focused or derived from probability theory,
economic behavior or dynamic programming. Here, we propose an approach to
define and implement dynamic market risk measures based on recursion and state
economy representation. The proposed approach is to be implementable and to
inherit properties from static market risk measures.Comment: 16 pages, 3 figure
R(p,q)- analogs of discrete distributions: general formalism and application
In this paper, we define and discuss - deformations of
basic univariate discrete distributions of the probability theory. We mainly
focus on binomial, Euler, P\'olya and inverse P\'olya distributions. We discuss
relevant deformed factorial moments of a random variable,
and establish associated expressions of mean and variance. Futhermore, we
derive a recursion relation for the probability distributions. Then, we apply
the same approach to build main distributional properties characterizing the
generalized Quesne quantum algebra, used in physics. Other known results
in the literature are also recovered as particular cases
Rough path recursions and diffusion approximations
In this article, we consider diffusion approximations for a general class of
stochastic recursions. Such recursions arise as models for population growth,
genetics, financial securities, multiplicative time series, numerical schemes
and MCMC algorithms. We make no particular probabilistic assumptions on the
type of noise appearing in these recursions. Thus, our technique is well suited
to recursions where the noise sequence is not a semi-martingale, even though
the limiting noise may be. Our main theorem assumes a weak limit theorem on the
noise process appearing in the random recursions and lifts it to diffusion
approximation for the recursion itself. To achieve this, we approximate the
recursion (pathwise) by the solution to a stochastic equation driven by
piecewise smooth paths; this can be thought of as a pathwise version of
backward error analysis for SDEs. We then identify the limit of this stochastic
equation, and hence the original recursion, using tools from rough path theory.
We provide several examples of diffusion approximations, both new and old, to
illustrate this technique.Comment: Published at http://dx.doi.org/10.1214/15-AAP1096 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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