354 research outputs found
Statistical Skorohod embedding problem and its generalizations
Given a L\'evy process , we consider the so-called statistical Skorohod
embedding problem of recovering the distribution of an independent random time
based on i.i.d. sample from Our approach is based on the genuine
use of the Mellin and Laplace transforms. We propose a consistent estimator for
the density of derive its convergence rates and prove their optimality. It
turns out that the convergence rates heavily depend on the decay of the Mellin
transform of We also consider the application of our results to the
problem of statistical inference for variance-mean mixture models and for
time-changed L\'evy processes
Statistical inference for generalized Ornstein-Uhlenbeck processes
In this paper, we consider the problem of statistical inference for
generalized Ornstein-Uhlenbeck processes of the type where is a
L{\'e}vy process. Our primal goal is to estimate the characteristics of the
L\'evy process from the low-frequency observations of the process
. We present a novel approach towards estimating the L{\'e}vy triplet of
which is based on the Mellin transform technique. It is shown that the
resulting estimates attain optimal minimax convergence rates. The suggested
algorithms are illustrated by numerical simulations.Comment: 32 pages. arXiv admin note: text overlap with arXiv:1312.473
Calculation of some determinants using the s-shifted factorial
Several determinants with gamma functions as elements are evaluated. This
kind of determinants are encountered in the computation of the probability
density of the determinant of random matrices. The s-shifted factorial is
defined as a generalization for non-negative integers of the power function,
the rising factorial (or Pochammer's symbol) and the falling factorial. It is a
special case of polynomial sequence of the binomial type studied in
combinatorics theory. In terms of the gamma function, an extension is defined
for negative integers and even complex values. Properties, mainly composition
laws and binomial formulae, are given. They are used to evaluate families of
generalized Vandermonde determinants with s-shifted factorials as elements,
instead of power functions.Comment: 25 pages; added section 5 for some examples of application
Dirichlet random walks
This article provides tools for the study of the Dirichlet random walk in
. By this we mean the random variable
where is Dirichlet distributed and where
are iid, uniformly distributed on the unit sphere of
and independent of In particular we compute explicitely in
a number of cases the distribution of Some of our results appear already
in the literature, in particular in the papers by G\'erard Le Ca\"{e}r (2010,
2011). In these cases, our proofs are much simpler from the original ones,
since we use a kind of Stieltjes transform of instead of the Laplace
transform: as a consequence the hypergeometric functions replace the Bessel
functions. A crucial ingredient is a particular case of the classical and non
trivial identity, true for : We extend these results to a study of
the limits of the Dirichlet random walks when the number of added terms goes to
infinity, interpreting the results in terms of an integral by a Dirichlet
process. We introduce the ideas of Dirichlet semigroups and of Dirichlet
infinite divisibility and characterize these infinite divisible distributions
in the sense of Dirichlet when they are concentrated on the unit ball of
{4mm}\noindent \textsc{Keywords:} Dirichlet processes, Stieltjes transforms,
random flight, distributions in a ball, hyperuniformity, infinite divisibility
in the sense of Dirichlet.
{4mm}\noindent \textsc{AMS classification}: 60D99, 60F99
A Stringy Mechanism for A Small Cosmological Constant
Based on the probability distributions of products of random variables, we
propose a simple stringy mechanism that prefers the meta-stable vacua with a
small cosmological constant. We state some relevant properties of the
probability distributions of functions of random variables. We then illustrate
the mechanism within the flux compactification models in Type IIB string
theory. As a result of the stringy dynamics, we argue that the generic
probability distribution for the meta-stable vacua typically peaks with a
divergent behavior at the zero value of the cosmological constant. However, its
suppression in the single modulus model studied here is modest.Comment: 36 pages, 8 figure
Completeness and Nonclassicality of Coherent States for Generalized Oscillator Algebras
The purposes of this work are (1) to show that the appropriate
generalizations of the oscillator algebra permit the construction of a wide set
of nonlinear coherent states in unified form; and (2) to clarify the likely
contradiction between the nonclassical properties of such nonlinear coherent
states and the possibility of finding a classical analog for them since they
are P-represented by a delta function. In (1) we prove that a class of
nonlinear coherent states can be constructed to satisfy a closure relation that
is expressed uniquely in terms of the Meijer G-function. This property
automatically defines the delta distribution as the P-representation of such
states. Then, in principle, there must be a classical analog for them. Among
other examples, we construct a family of nonlinear coherent states for a
representation of the su(1,1) Lie algebra that is realized as a deformation of
the oscillator algebra. In (2), we use a beam splitter to show that the
nonlinear coherent states exhibit properties like anti-bunching that prohibit a
classical description for them. We also show that these states lack second
order coherence. That is, although the P-representation of the nonlinear
coherent states is a delta function, they are not full coherent. Therefore, the
systems associated with the generalized oscillator algebras cannot be
considered `classical' in the context of the quantum theory of optical
coherence.Comment: 26 pages, 10 figures, minor changes, misprints correcte
From duality to determinants for q-TASEP and ASEP
We prove duality relations for two interacting particle systems: the
-deformed totally asymmetric simple exclusion process (-TASEP) and the
asymmetric simple exclusion process (ASEP). Expectations of the duality
functionals correspond to certain joint moments of particle locations or
integrated currents, respectively. Duality implies that they solve systems of
ODEs. These systems are integrable and for particular step and half-stationary
initial data we use a nested contour integral ansatz to provide explicit
formulas for the systems' solutions, and hence also the moments. We form
Laplace transform-like generating functions of these moments and via residue
calculus we compute two different types of Fredholm determinant formulas for
such generating functions. For ASEP, the first type of formula is new and
readily lends itself to asymptotic analysis (as necessary to reprove GUE
Tracy--Widom distribution fluctuations for ASEP), while the second type of
formula is recognizable as closely related to Tracy and Widom's ASEP formula
[Comm. Math. Phys. 279 (2008) 815--844, J. Stat. Phys. 132 (2008) 291--300,
Comm. Math. Phys. 290 (2009) 129--154, J. Stat. Phys. 140 (2010) 619--634]. For
-TASEP, both formulas coincide with those computed via Borodin and Corwin's
Macdonald processes [Probab. Theory Related Fields (2014) 158 225--400]. Both
-TASEP and ASEP have limit transitions to the free energy of the continuum
directed polymer, the logarithm of the solution of the stochastic heat equation
or the Hopf--Cole solution to the Kardar--Parisi--Zhang equation. Thus,
-TASEP and ASEP are integrable discretizations of these continuum objects;
the systems of ODEs associated to their dualities are deformed discrete quantum
delta Bose gases; and the procedure through which we pass from expectations of
their duality functionals to characterizing generating functions is a rigorous
version of the replica trick in physics.Comment: Published in at http://dx.doi.org/10.1214/13-AOP868 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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