arXiv subjects

Jacek Wesolowski

Publications and source records attributed to Jacek Wesolowski.

At least 19 recordsLinked to original sources

arXiv2026

Stationary measures for log-gamma polymer on a strip and in half-space

We study stationary measures of the log-gamma polymer on a finite diagonal strip and in a half-space. We establish the phase diagram for the stationary measure on the strip. To this end we develop a representation of {the Laplace transform of} this stationary measure by independent $\mathrm{Beta}_{II}$ random variables. We also present an analytic approach that extends Barraquand's contour integral representation of the Laplace transform. We prove that, as the strip width tends to infinity, the stationary measure of the log-gamma polymer on the strip converges to the stationary measure of the half-space log-gamma polymer. Finally, we derive a contour integral formula for the Laplace transform of the stationary measure of the half-space log-gamma polymer.

math.PR↗

arXiv2025

Free Askey--Wilson functionals and geometric last passage percolation on a strip

Barraquand, Corwin, and Yang arXiv:2306.05983 established that geometric last passage percolation (LPP) on a strip of $\mathbb{Z}^2$ has a unique stationary measure. Building on this, Barraquand arXiv:2409.08927 derived explicit contour integral formulas for the model's multipoint probability generating function. In this paper, we introduce free Askey--Wilson functionals and use them to extend these generating function formulas. Our framework yields explicit expressions valid over a broader range of boundary parameters than previously accessible. This generalization allows us to determine the full phase diagram that characterizes how the large-scale asymptotics of the stationary measure depend on the boundary conditions. In addition, we prove a Poisson approximation for the stationary measure when the parameters vary with the strip width.

math.PR↗

arXiv2025

Limits of Random Motzkin paths with KPZ related asymptotics

We study Motzkin paths of length $L$ with general weights on the edges and end points. We investigate the limit behavior of the initial and final segments of the random Motzkin path viewed as a pair of processes starting from each of the two end points as $L$ becomes large. We then study macroscopic limits of the resulting processes, where in two different regimes we obtain Markov processes that appeared in the description of the stationary measure for the KPZ equation on the half line and of conjectural stationary measure of the hypothetical KPZ fixed point on the half line. Our results rely on the behavior of the Al-Salam--Chihara polynomials in the neighbourhood of the upper end of their orthogonality interval and on the limiting properties of the $q$-Pochhammer and $q$-Gamma functions as $q\nearrow 1$.

math.PR↗

arXiv2024

Pitman's discrete $2M-X$ theorem for arbitrary initial laws and continuous time limits

We discuss Pitman's representation of a Markov process, which serves as a discrete analog to the Bessel 3D process starting at time 0 from an arbitrary initial law. This representation involves maxima of lazy simple random walks and an auxiliary independent random variable. The law of the auxiliary random variable is explicitly related to the initial law of the Markov process. The proof is kept at an elementary level and relies on a reconstruction formula for the generalized Pitman transform. We then use continuous-time limits to shed additional light on the relation between two representations of the Bessel 3D process that appeared in the description of the stationary measure of the KPZ fixed point on the half-line, as proposed by Barraquand and Le Doussal (2022).

math.PR↗

arXiv2024

Askey-Wilson signed measures and open ASEP in the shock region

We introduce a family of multi-dimensional Askey-Wilson signed measures. We offer an explicit description of the stationary measure of the open asymmetric simple exclusion process (ASEP) in the full phase diagram, in terms of integrations with respect to these Askey-Wilson signed measures. Using our description, we provide a rigorous derivation of the density profile and limit fluctuations of open ASEP in the entire shock region, including the high and low density phases as well as the coexistence line. This in particular confirms the existing physics postulations of the density profile.

math.PR↗

arXiv2022

Markov processes related to the stationary measure for the open KPZ equation

We provide a probabilistic description of the stationary measures for the open KPZ on the spatial interval $[0,1]$ in terms of a Markov process $Y$, which is a Doob's $h$ transform of the Brownian motion killed at an exponential rate. Our work builds on a recent formula of Corwin and Knizel which expresses the multipoint Laplace transform of the stationary solution of the open KPZ in terms of another Markov process $\mathbb T$: the continuous dual Hahn process with Laplace variables taking on the role of time-points in the process. The core of our approach is to prove that the Laplace transforms of the finite dimensional distributions of $Y$ and $\mathbb T$ are equal when the time parameters of one process become the Laplace variables of the other process and vice versa.

math.PR↗

arXiv2022

From the asymmetric simple exclusion processes to the stationary measures of the KPZ fixed point on an interval

Barraquand and Le~Doussal introduced a family of stationary measures for the (conjectural) KPZ fixed point on an interval with Neumann boundary conditions, and predicted that they arise as scaling limit of stationary measures of all models in the KPZ universality class on an interval. In this paper, we show that the stationary measures for KPZ fixed point on an interval arise as the scaling limits of the height increment processes for the open asymmetric simple exclusion process in the steady state, with parameters changing appropriately as the size of the system tends to infinity.

math.PR↗

arXiv2018

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.

math.PR↗

arXiv2015

Infinitesimal generators for a class of polynomial processes

We study the infinitesimal generators of evolutions of linear mappings on the space of polynomials, which correspond to a special class of Markov processes with polynomial regressions called quadratic harnesses. We relate the infinitesimal generator to the unique solution of a certain commutation equation, and we use the commutation equation to find an explicit formula for the infinitesimal generator of free quadratic harnesses.

math.PR↗

arXiv2015

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.

math.ST↗

arXiv2015

Regression version of the Matsumoto-Yor type characterization of the gamma and Kummer distributions

In this paper we study a Matsumoto-Yor type property for the gamma and Kummer inde- pendent variables discovered in Koudou and Vallois (2012). We prove that constancy of regressions of U = (1 + 1/(X + Y ))=(1 + 1/X) given V = X + Y and of 1/U given V , where X and Y are indepen- dent and positive random variables, characterizes the gamma and Kummer distributions. This result completes characterizations by independence of U and V obtained, under smoothness assumptions for densities, in Koudou and Vallois (2011, 2012). Since we work with differential equations for the Laplace transforms, no density assumptions are needed.

math.PR↗

arXiv2014

A new prior for the discrete DAG models with a restricted set of directions

In this paper, we first develop a new family of conjugate prior distributions for the cell parameters of discrete graphical models Markov with respect to a set P of moral directed acyclic graphs with skeleton a given decomposable graph G. Such families arise when the set of conditional independences between discrete variables is given and can be represented by a decomposable graph and additionally, the direction of certain edges is imposed by the practitioner. This family, which we call the P-Dirichlet, is a generalization of the hyper Dirichlet given in Dawid and Lauritzen (1993): it keeps the strong directed hyper Markov property for every DAG in P but increases the flexibility in the choice of its parameters, i.e. the hyper parameters. Our second contribution is a characterization of the P-Dirichlet, which yields, as a corollary, a characterization of the hyper Dirichlet and a characterization of the Dirichlet also. Like that given by Geiger and Heckerman (1997), our characterization of the Dirichlet is based on local and global independence of the probability parameters but we need not make the assumption of the existence of a positive density function. We use the method of moments for our proofs.

math.ST↗

arXiv2014

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.

math.ST↗

arXiv2013

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.

math.PR↗

arXiv2013

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 $U$, $V$ are independent real random variables, such that $E(V(1-U)|UV)$ and $E(V^2(1-U)^2|UV)$ are non-random then $V$ has a gamma distribution and $U$ has a beta distribution.

math.OA↗

arXiv2013

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.

math.PR↗

arXiv2012

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.

math.PR↗

arXiv2011

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.

math.PR↗