1,754 research outputs found
Integral expressions for Hilbert-type infinite multilinear form and related multiple Hurwitz-Lerch Zeta functions
The article deals with different kinds integral expressions concerning multiple Hurwitz-Lerch Zeta function (introduced originally by Barnes ), Hilbert-type infinite multilinear form and its power series extension. Here Laplace integral forms and multiple Mellin-Barnes type integral representation are derived for these special functions. As a special cases of our investigations we deduce the integral expressions for the Matsumoto's multiple Mordell-Tornheim Zeta function, that is, for Tornheim's double sum i.e. Mordell-Witten Zeta, for the multiple Hurwitz Zeta and for the multiple Hurwitz-Euler Eta function, recently studied by Choi and Srivastava
Purity-bounded uncertainty relations in multidimensional space -- generalized purity
Uncertainty relations for mixed quantum states (precisely, purity-bounded
position-momentum relations, developed by Bastiaans and then by Man'ko and
Dodonov) are studied in general multi-dimensional case. An expression for
family of mixed states at the lower bound of uncertainty relation is obtained.
It is shown, that in case of entropy-bounded uncertainty relations, lower-bound
state is thermal, and a transition from one-dimensional problem to
multi-dimensional one is trivial. Results of numerical calculation of the
relation lower bound for different types of generalized purity are presented.
Analytical expressions for general purity-bounded relations for highly mixed
states are obtained.Comment: 12 pages, 2 figures. draft version, to appear in J. Phys. A Partially
based on a poster "Multidimensional uncertainty relations for states with
given generalized purity" presented on X Intl. Conf. on Quantum Optics'2004
(Minsk, Belarus, May 30 -- June 3, 2004) More actual report is to be
presented on ICSSUR-2005, Besan\c{c}on, France and on EQEC'05, Munich. V. 5:
amended article after referees' remark
Universality of free homogeneous sums in every dimension
We prove a general multidimensional invariance principle for a family of
U-statistics based on freely independent non-commutative random variables of
the type , where is the -th Chebyshev polynomial and is
a standard semicircular element on a fixed -probability space. As a
consequence, we deduce that homogeneous sums based on random variables of this
type are universal with respect to both semicircular and free Poisson
approximations.
Our results are stated in a general multidimensional setting and can be seen
as a genuine extension of some recent findings by Deya and Nourdin; our
techniques are based on the combination of the free Lindeberg method and the
Fourth moment Theorem
Convergence towards linear combinations of chi-squared random variables: a Malliavin-based approach
We investigate the problem of finding necessary and sufficient conditions for
convergence in distribution towards a general finite linear combination of
independent chi-squared random variables, within the framework of random
objects living on a fixed Gaussian space. Using a recent representation of
cumulants in terms of the Malliavin calculus operators (introduced
by Nourdin and Peccati in \cite{n-pe-3}), we provide conditions that apply to
random variables living in a finite sum of Wiener chaoses. As an important
by-product of our analysis, we shall derive a new proof and a new
interpretation of a recent finding by Nourdin and Poly \cite{n-po-1},
concerning the limiting behaviour of random variables living in a Wiener chaos
of order two. Our analysis contributes to a fertile line of research, that
originates from questions raised by Marc Yor, in the framework of limit
theorems for non-linear functionals of Brownian local times
On the reproducing kernel Hilbert spaces associated with the fractional and bi-fractional Brownian motions
We present decompositions of various positive kernels as integrals or sums of
positive kernels. Within this framework we study the reproducing kernel Hilbert
spaces associated with the fractional and bi-fractional Brownian motions. As a
tool, we define a new function of two complex variables, which is a natural
generalization of the classical Gamma function for the setting we conside
Rosenblatt distribution subordinated to gaussian random fields with long-range dependence
The Karhunen-Lo\`eve expansion and the Fredholm determinant formula are used
to derive an asymptotic Rosenblatt-type distribution of a sequence of integrals
of quadratic functions of Gaussian stationary random fields on R^d displaying
long-range dependence. This distribution reduces to the usual Rosenblatt
distribution when d=1. Several properties of this new distribution are
obtained. Specifically, its series representation in terms of independent
chi-squared random variables is given, the asymptotic behavior of the
eigenvalues, its L\`evy-Khintchine representation, as well as its membership to
the Thorin subclass of self-decomposable distributions. The existence and
boundedness of its probability density is then a direct consequence.Comment: This paper has 40 pages and it has already been submitte
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