55 research outputs found
Asymmetric Simple Exclusion Process with open boundaries and Quadratic Harnesses
We show that the joint probability generating function of the stationary
measure of a finite state asymmetric exclusion process with open boundaries can
be expressed in terms of joint moments of Markov processes called quadratic
harnesses. We use our representation to prove the large deviations principle
for the total number of particles in the system. We use the generator of the
Markov process to show how explicit formulas for the average occupancy of a
site arise for special choices of parameters. We also give similar
representations for limits of stationary measures as the number of sites tends
to infinity.Comment: Corrected more typo
Dual Lukacs regressions for non-commutative variables
Dual Lukacs type characterizations of random variables in free probability
are studied here. First, we develop a freeness property satisfied by Lukacs
type transformations of free-Poisson and free-Binomial non-commutative
variables which are free. Second, we give a characterization of non-commutative
free-Poisson and free-Binomial variables by properties of first two conditional
moments, which mimic Lukacs type assumptions known from classical probability.
More precisely, our result is a non-commutative version of the following result
known in classical probability: if , are independent real random
variables, such that and are non-random then
has a gamma distribution and has a beta distribution
Classical bi-Poisson process: an invertible quadratic harness
We give an elementary construction of a time-invertible Markov process which
is discrete except at one instance. The process is one of the quadratic
harnesses studied in our previous papers and can be regarded as a random joint
of two independent Poisson processes
An eigenproblem approach to optimal equal-precision sample allocation in subpopulations
Allocation of samples in stratified and/or multistage sampling is one of the
central issues of sampling theory. In a survey of a population often the
constraints for precision of estimators of subpopulations parameters have to be
taken care of during the allocation of the sample. Such issues are often solved
with mathematical programming procedures. In many situations it is desirable to
allocate the sample, in a way which forces the precision of estimates at the
subpopulations level to be both: optimal and identical, while the constraints
of the total (expected) size of the sample (or samples, in two-stage sampling)
are imposed. Here our main concern is related to two-stage sampling schemes. We
show that such problem in a wide class of sampling plans has an elegant
mathematical and computational solution. This is done due to a suitable
definition of the optimization problem, which enables to solve it through a
linear algebra setting involving eigenvalues and eigenvectors of matrices
defined in terms of some population quantities. As a final result we obtain a
very simple and relatively universal method for calculating the subpopulation
optimal and equal-precision allocation which is based on one of the most
standard algorithms of linear algebra (available e.g. in R software).
Theoretical solutions are illustrated through a numerical example based on the
Labour Force Survey. Finally, we would like to stress that the method we
describe, allows to accommodate quite automatically for different levels of
precision priority for subpopulations
Infinitesimal generators of q-Meixner processes
We show that the weak infinitesimal generator of a class of Markov processes
acts on bounded continuous functions with bounded continuous second derivative
as a singular integral with respect to the orthogonality measure of the
explicit family of polynomials
Quadratic Harnesses, q-commutations, and orthogonal martingale polynomials
We introduce the quadratic harness condition and show that integrable
quadratic harnesses have orthogonal martingale polynomials with a three step
recurrence that satisfies a q-commutation relation. This implies that quadratic
harnesses are essentially determined uniquely by five numerical constants.
Explicit recurrences for the orthogonal martingale polynomials are derived in
several cases of interest
The Lukacs theorem and the Olkin-Baker equation
The Olkin-Baker functional equation is closely related to the celebrated
Lukacs characterization of the gamma distribution. Its deeper understanding is
essential to settle a challenging question of multivariate extensions of the
Lukacs theorem. In this paper, first, we provide a new approach to the additive
Olkin-Baker equation which holds almost everywhere on (0,\infinity)^2 (with
respect to the Lebesgue measure on R^2) under measurability assumption. Second,
this new approach is adapted to the case when unknown functions are allowed to
be non-measurable and the complete solution is given in such a general case.
Third, the Olkin-Baker equation holding outside of a set from proper linearly
invariant ideal of subsets of R^2 is considered
Stitching pairs of Levy processes into harnesses
We consider natural exponential families of Levy processes with randomized
parameter. Such processes are Markov, and under suitable assumptions, pairs of
such processes with shared randomization can be stitched together into a single
harness. The stitching consists of deterministic reparametrization of the time
for both processes, so that they run on adjacent time intervals, and of the
choice of the appropriate law at the boundary.
Processes in the Levy-Meixner class have an additional property that they are
quadratic harnesses, and in this case stitching constructions produce quadratic
harnesses
Exploring recursion for optimal estimators under cascade rotation
We are concerned with optimal linear estimation of means on subsequent
occasions under sample rotation where evolution of samples in time is designed
through a cascade pattern. It has been known since the seminal paper of
Patterson (1950) that when the units are not allowed to return to the sample
after leaving it for certain period (there are no gaps in the rotation
pattern), one step recursion for optimal estimator holds. However, in some
important real surveys, e.g. Current Population Survey in the US or Labour
Force Survey in many countries in Europe, units return to the sample after
being absent in the sample for several occasions (there are gaps in rotation
patterns). In such situations difficulty of the question of the form of the
recurrence for optimal estimator increases drastically. This issue has not been
resolved yet. Instead alternative sub-optimal approaches were developed, as
K-composite estimation (see e.g. Hansen et al. (1955)), AK-composite estimation
(see e.g. Gurney and Daly (1965) or time series approach (see e.g. Binder and
Hidiroglou (1988)).
In the present paper we overcome this long-standing difficulty, that is, we
present analytical recursion formulas for the optimal linear estimator of the
mean for schemes with gaps in rotation patterns. It is achieved under some
technical conditions: ASSUMPTION I and ASSUMPTION II (numerical experiments
suggest that these assumptions might be universally satisfied). To attain the
goal we develop an algebraic operator approach which allows to reduce the
problem of recursion for the optimal linear estimator to two issues: (1)
localization of roots (possibly complex) of a polynomial Q_p defined in terms
of the rotation pattern (Q_p happens to be conveniently expressed through
Chebyshev polynomials of the first kind), (2) rank of a matrix S defined in
terms of the rotation pattern and the roots of the polynomial Q_p
Asymptotic normality through factorial cumulants and partitions identities
In the paper we develop an approach to asymptotic normality through factorial
cumulants. Factorial cumulants arise in the same manner from factorial moments,
as do (ordinary) cumulants from (ordinary) moments. Another tool we exploit is
a new identity for "moments" of partitions of numbers. The general limiting
result is then used to (re-)derive asymptotic normality for several models
including classical discrete distributions, occupancy problems in some
generalized allocation schemes and two models related to negative multinomial
distribution
- …