13,381 research outputs found
An asymptotic relationship between coupling methods for stochastically modeled population processes
This paper is concerned with elucidating a relationship between two common
coupling methods for the continuous time Markov chain models utilized in the
cell biology literature. The couplings considered here are primarily used in a
computational framework by providing reductions in variance for different Monte
Carlo estimators, thereby allowing for significantly more accurate results for
a fixed amount of computational time. Common applications of the couplings
include the estimation of parametric sensitivities via finite difference
methods and the estimation of expectations via multi-level Monte Carlo
algorithms. While a number of coupling strategies have been proposed for the
models considered here, and a number of articles have experimentally compared
the different strategies, to date there has been no mathematical analysis
describing the connections between them. Such analyses are critical in order to
determine the best use for each. In the current paper, we show a connection
between the common reaction path (CRP) method and the split coupling (SC)
method, which is termed coupled finite differences (CFD) in the parametric
sensitivities literature. In particular, we show that the two couplings are
both limits of a third coupling strategy we call the "local-CRP" coupling, with
the split coupling method arising as a key parameter goes to infinity, and the
common reaction path coupling arising as the same parameter goes to zero. The
analysis helps explain why the split coupling method often provides a lower
variance than does the common reaction path method, a fact previously shown
experimentally.Comment: Edited Section 4.
Stein meets Malliavin in normal approximation
Stein's method is a method of probability approximation which hinges on the
solution of a functional equation. For normal approximation the functional
equation is a first order differential equation. Malliavin calculus is an
infinite-dimensional differential calculus whose operators act on functionals
of general Gaussian processes. Nourdin and Peccati (2009) established a
fundamental connection between Stein's method for normal approximation and
Malliavin calculus through integration by parts. This connection is exploited
to obtain error bounds in total variation in central limit theorems for
functionals of general Gaussian processes. Of particular interest is the fourth
moment theorem which provides error bounds of the order
in the central limit theorem for elements
of Wiener chaos of any fixed order such that
. This paper is an exposition of the work of Nourdin and
Peccati with a brief introduction to Stein's method and Malliavin calculus. It
is based on a lecture delivered at the Annual Meeting of the Vietnam Institute
for Advanced Study in Mathematics in July 2014.Comment: arXiv admin note: text overlap with arXiv:1404.478
First passage percolation on the Newman-Watts small world model
The Newman-Watts model is given by taking a cycle graph of n vertices and
then adding each possible edge with probability
for some constant. In this paper we add i.i.d. exponential
edge weights to this graph, and investigate typical distances in the
corresponding random metric space given by the least weight paths between
vertices. We show that typical distances grow as for a
and determine the distribution of smaller order terms in terms of
limits of branching process random variables. We prove that the number of edges
along the shortest weight path follows a Central Limit Theorem, and show that
in a corresponding epidemic spread model the fraction of infected vertices
follows a deterministic curve with a random shift.Comment: 29 pages, 4 figure
Percolation and Connectivity in the Intrinsically Secure Communications Graph
The ability to exchange secret information is critical to many commercial,
governmental, and military networks. The intrinsically secure communications
graph (iS-graph) is a random graph which describes the connections that can be
securely established over a large-scale network, by exploiting the physical
properties of the wireless medium. This paper aims to characterize the global
properties of the iS-graph in terms of: (i) percolation on the infinite plane,
and (ii) full connectivity on a finite region. First, for the Poisson iS-graph
defined on the infinite plane, the existence of a phase transition is proven,
whereby an unbounded component of connected nodes suddenly arises as the
density of legitimate nodes is increased. This shows that long-range secure
communication is still possible in the presence of eavesdroppers. Second, full
connectivity on a finite region of the Poisson iS-graph is considered. The
exact asymptotic behavior of full connectivity in the limit of a large density
of legitimate nodes is characterized. Then, simple, explicit expressions are
derived in order to closely approximate the probability of full connectivity
for a finite density of legitimate nodes. The results help clarify how the
presence of eavesdroppers can compromise long-range secure communication.Comment: Submitted for journal publicatio
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