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In this paper the relations between independence preserving (IP) involutions and reversible Markov kernels are investigated. We introduce an involutive augmentation H = (f, g_f) of a measurable function f and relate the IP property of H to f-generated reversible Markov kernels. Various examples appeared in the literature are presented as particular cases of the construction. In particular, we prove that the IP property generated by the (reversible) Markov kernel of random walk with a reflecting barrier at the origin characterizes geometric-type laws, Comment: 14 pages

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 $q\nearrow 1$.

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., Comment: 25 pages. v3: Minor edits

Quadratic harnesses are time-inhomogeneous Markov polynomial processes with linear conditional expectations and quadratic conditional variances with respect to the past-future filtrations. Typically they are determined by five numerical constants hidden in the form of conditional variances. In this paper we derive infinitesimal generators of such processes, extending previously known results. The infinitesimal generators are identified through a solution of a q-commutation equation in the algebra Q of infinite sequences of polynomials in one variable. The solution is a special element in Q, whose coordinates satisfy a three-term recurrence and thus define a system of orthogonal polynomials. It turns out that the respective orthogonality measure uniquely determines the infinitesimal generator (acting on polynomials or bounded functions with bounded continuous second derivative) as an integro-differential operator with the explicit kernel, where the integration is with respect to this measure.

We extend Pitman's representation of a discrete analog of the Bessel 3d process that starts at 0 to arbitrary initial laws. The representation is in terms of maxima of simple random walks and can be interpreted as a random walk conditioned to be above a random level. As an application we establish continuous time limit, which in particular yields convergence to the non-Gaussian component of a stationary measure of the KPZ fixed point on the half-line as proposed in Barraquand and Le Doussal (2022).

The optimum sample allocation in stratified sampling is one of the basic issues of survey methodology. It is a procedure of dividing the overall sample size into strata sample sizes in such a way that for given sampling designs in strata the variance of the stratified $\pi$ estimator of the population total (or mean) for a given study variable assumes its minimum. In this work, we consider the optimum allocation of a sample, under lower and upper bounds imposed jointly on sample sizes in strata. We are concerned with the variance function of some generic form that, in particular, covers the case of the simple random sampling without replacement in strata. The goal of this paper is twofold. First, we establish (using the Karush-Kuhn-Tucker conditions) a generic form of the optimal solution, the so-called optimality conditions. Second, based on the established optimality conditions, we derive an efficient recursive algorithm, named RNABOX, which solves the allocation problem under study. The RNABOX can be viewed as a generalization of the classical recursive Neyman allocation algorithm, a popular tool for optimum allocation when only upper bounds are imposed on sample strata-sizes. We implement RNABOX in R as a part of our package stratallo which is available from the Comprehensive R Archive Network (CRAN) repository.

Bayesian statistical graphical models are typically either continuous and parametric (Gaussian, parameterized by the graph-dependent precision matrix with Wishart-type priors) or discrete and non-parametric (with graph-dependent structure of probabilities of cells and Dirichlet-type priors). We propose to break this dichotomy by introducing two discrete parametric graphical models on finite decomposable graphs: the graph negative multinomial and the graph multinomial distributions. These models interpolate between the product of univariate negative binomial laws and the negative multinomial distribution, and between the product of binomial laws and the multinomial distribution, respectively. We derive their Markov decomposition and present related probabilistic models representations. We also introduce graphical versions of the Dirichlet distribution and inverted Dirichlet distribution, which serve as conjugate priors for the two discrete graphical Markov models. We derive explicit normalizing constants for both graphical Dirichlet laws and demonstrate that their independence structure (a graphical version of neutrality) yields a strong hyper Markov property for both Bayesian models. We also provide characterization theorems for graphical Dirichlet laws via strong hyper Markov property. Finally, we develop a model selection procedure for the Bayesian graphical negative multinomial model with respective Dirichlet-type priors., Comment: 40 pages

We prove that if $X,Y$ are positive, independent, non-Dirac random variables and if for $\alpha,\beta\ge 0$, $\alpha\neq \beta$, $$ \psi_{\alpha,\beta}(x,y)=\left(y\,\tfrac{1+\beta(x+y)}{1+\alpha x+\beta y},\;x\,\tfrac{1+\alpha(x+y)}{1+\alpha x+\beta y}\right), $$ then the random variables $U$ and $V$ defined by $(U,V)=\psi_{\alpha,\beta}(X,Y)$ are independent if and only if $X$ and $Y$ follow Kummer distributions with suitably related parameters. In other words, any invariant measure for a lattice recursion model governed by $\psi_{\alpha,\beta}$ in the scheme introduced by Croydon and Sasada in \cite{CS2020} is necessarily a product measure with Kummer marginals. The result extends earlier characterizations of Kummer and gamma laws by independence of $$ U=\tfrac{Y}{1+X}\quad\mbox{and}\quad V= X\left(1+\tfrac{Y}{1+X}\right), $$ which corresponds to the case of $\psi_{1,0}$. We also show that this independence property of Kummer laws covers, as limiting cases, several independence models known in the literature: the Lukacs, the Kummer-Gamma, the Matsumoto-Yor and the discrete Korteweg de Vries models.

If $\alpha,\beta>0$ are distinct and if $A$ and $B$ are independent non-degenerate positive random variables such that $$S=\tfrac{1}{B}\,\tfrac{\beta A+B}{\alpha A+B}\quad \mbox{and}\quad T=\tfrac{1}{A}\,\tfrac{\beta A+B}{\alpha A+B} $$ are independent, we prove that this happens if and only if the $A$ and $B$ have generalized inverse Gaussian distributions with suitable parameters. Essentially, this has already been proved in Bao and Noack (2021) with supplementary hypothesis on existence of smooth densities. The sources of these questions are an observation about independence properties of the exponential Brownian motion due to Matsumoto and Yor (2001) and a recent work of Croydon and Sasada (2000) on random recursion models rooted in the discrete Korteweg - de Vries equation, where the above result was conjectured. We also extend the direct result to random matrices proving that a matrix variate analogue of the above independence property is satisfied by independent matrix-variate GIG variables. The question of characterization of GIG random matrices through this independence property remains open., Comment: 17 pages

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.

We point out to a connection between a problem of invariance of power series families of probability distributions under binomial thinning and functional equations which generalize both the Cauchy and an additive form of the Go{\l}ab-Schinzel equation. We solve these equations in several settings with no or mild regularity assumptions imposed on unknown functions.

We derive a formula for the optimal sample allocation in a general stratified scheme under upper bounds on the sample strata-sizes. Such a general scheme includes SRSWOR within strata as a special case. The solution is given in terms of V-allocation with V being the set of take-all strata. We use V-allocation to give a formal proof of optimality of the popular recursive Neyman algorithm, rNa. This approach is convenient also for a quick proof of optimality of the algorithm of Stenger and Gabler (2005), SGa, as well as of its modification, coma, we propose here. Finally, we compare running times of rNa, SGa and coma. Ready-to-use R-implementations of these algorithms are available on CRAN repository at https://cran.r-project.org/web/packages/stratallo.

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., Comment: Expanded version, typo in (6.6) corrected

We introduce one-way flows in near algebras and two-way flows in double near algebras with two interrelated multiplications. We establish parametric representations of the one-way and two-way flows in terms of a single element of the algebra that we call a flow generator. We indicate probabilistic applications of the one-way flows to a study of polynomial stochastic processes. We apply our results on the two-way flows to harnesses and quadratic harnesses in probability theory, generalizing some previous results.

We use here a recent idea of studying functions of free random variables using Boolean cumulants. We develop idea of explicit calculations of conditional expectation using Boolean cumulants. We demonstrate Boolean cumulants approach allows to calculate explicitly some conditional expectations of functions in free random variables. We present how Boolean cumulants together with subordination simplify proofs of some results which are known in research literature.

Consider a number, finite or not, of urns each with fixed capacity $r$ and balls randomly distributed among them. An overflow is the number of balls that are assigned to urns that already contain $r$ balls. When $r=1$, using analytic methods, Hwang and Janson gave conditions under which the overflow (which in this case is just the number of balls landing in non--empty urns) has an asymptotically Poisson distribution as the number of balls grows to infinity. Our aim here is to systematically study the asymptotics of the overflow in general situation, i.~e. for arbitrary $r$. In particular, we provide sufficient conditions for both Poissonian and normal asymptotics for general $r$, thus extending Hwang--Janson's work. Our approach relies on purely probabilistic methods., Comment: 4figures

In this paper we are interested in the joint distribution of two order statistics from overlapping samples. We give an explicit formula for the distribution of such a pair of random variables under the assumption that the parent distribution is absolutely continuous (with respect to the Lebesgue measure on the real line). The distribution is identified through the form of the density with respect to a measure which is a sum of the bivariate Lebesgue measure on $\R^2$ and the univariate Lebesgue measure on the diagonal $\{(x,x):\,x\in\R\}$. We are also interested in the question to what extent conditional expectation of one of such order statistic given another determines the parent distribution. In particular, we provide a new characterization by linearity of regression of an order statistic from the extended sample given the one from the original sample, special case of which solves a problem explicitly stated in the literature. It appears that to describe the correct parent distribution it is convenient to use quantile density functions. In several other cases of regressions of order statistics we provide new results regarding uniqueness of the distribution in the sample. Nevertheless the general question of identifiability of the parent distribution by regression of order statistics from overlapping samples remains open.

In Sabot and Tarr\`es (2015), the authors have explicitly computed the integral $$STZ_n=\int \exp( -\langle x,y\rangle)(\det M_x)^{-1/2}dx$$ where $M_x$ is a symmetric matrix of order $n$ with fixed non positive off-diagonal coefficients and with diagonal $(2x_1,\ldots,2x_n)$. The domain of integration is the part of $\mathbb{R}^n$ for which $M_x$ is positive definite. We calculate more generally for $ b_1\geq 0,\ldots b_n\geq 0$ the integral $$GSTZ_n=\int \exp \left(-\langle x,y\rangle-\frac{1}{2}b^*M_x^{-1}b\right)(\det M_x)^{-1/2}dx,$$ we show that it leads to a natural family of distributions in $\mathbb{R}^n$, called the $GSTZ_n$ probability laws. This family is stable by marginalization and by conditioning, and it has number of properties which are multivariate versions of familiar properties of univariate reciprocal inverse Gaussian distribution. We also show that if the graph with the set of vertices $V=\{1,\ldots,n\}$ and the set $E$ of edges $\{i,j\}'$ s of non zero entries of $M_x$ is a tree, then the integral $$\int \exp( -\langle x,y\rangle)(\det M_x)^{q-1}dx$$ where $q>0,$ is computable in terms of the MacDonald function $K_q.$, Comment: 17 pages

If $X$ and $Y$ are independent random variables with distributions $\mu$ and $\nu$ then $U=\psi(X,Y)$ and $V=\phi(X,Y)$ are also independent for some $\psi$ and $\phi$. Properties of this type are known for many important probability distributions $\mu$ and $\nu$. Also related characterization questions have been widely investigated: Let $X$ and $Y$ be independent and let $U$ and $V$ be independent. Are the distributions of $X$ and $Y$ $\mu$ and $\nu$, respectively? Recently two new properties and characterizations of this kind involving the Kummer distribution appeared in the literature. For independent $X$ and $Y$ with gamma and Kummer distributions Koudou and Vallois observed that $U=(1+(X+Y)^{-1})/(1+X^{-1})$ and $V=X+Y$ are also independent, and Hamza and Vallois observed that $U=Y/(1+X)$ and $V=X(1+Y/(1+X))$ are independent. In 2011 and 2012 Koudou, Vallois characterizations related to the first property were proved, while the characterizations in the second setting have been recently given in Piliszek, Weso{\l}owski (2016). In both cases technical assumptions on smoothness properties of densities of $X$ and $Y$ were needed. In 2015, the assumption of independence of $U$ and $V$ in the first setting was weakened to constancy of regressions of $U$ and $U^{-1}$ given $V$ with no density assumptions. However, the additional assumption $\mathbb{E} X^{-1}<\infty$ was introduced. In the present paper we provide a complete answer to the characterization question in both settings without any additional technical assumptions regarding smoothness or existence of moments. The approach is, first, via characterizations exploiting some conditions imposed on regressions of $U$ given $V$, which are weaker than independence, but for which moment assumptions are necessary. Second, using a technique of change of measure we show that the moment assumptions can be avoided., Comment: 12 pages

We consider distributional limit of the Pearson chi-square statistic when the number of classes m increases with the sample size n in such way that $n/\sqrt{m} \to {\lambda}$. Under mild moment conditions, the limit is Gaussian for ${\lambda} = \infty$, Poisson for finite ${\lambda} > 0$, and degenerate for ${\lambda} = 0$.

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 typos

Since many years characterization of distribution by linearity of regression of non-adjacent weak records E(W_{i+s}|W_i) = \beta_1 W_i+\beta_0 for discrete observations has been known to be a difficult question. Lopez- Blazquez (2004) proposed an interesting idea of reducing it to the adjacent case and claimed to have the characterization problem completely solved. We will explain that, unfortunately, there is a major aw in the proof given in that paper. This aw is related to fact that in some situations the operator responsible for reduction of the non-adjacent case to the adjacent one is not injective. The operator is trivially injective when 0<\beta_1<1. We show that when \beta_1>=1 the operator is injective when s = 2, 3, 4. Therefore in these cases the method proposed by Lopez-Blazquez is valid. We also show that the operator is not injective when \beta_1 >=1 and s >= 5. Consequently, in this case the reduction methodology does not work and thus the characterization problem remains open.

The paper develops a rather unexpected parallel to the multivariate Matsumoto--Yor (MY) property on trees considered in \cite{MW04}. The parallel concerns a multivariate version of the Kummer distribution, which is generated by a tree. Given a tree of size $p$, we direct it by choosing a vertex, say $r$, as a root. With such a directed tree we associate a map $\Phi_r$. For a random vector ${\bf S}$ having a $p$-variate tree-Kummer distribution and any root $r$, we prove that $\Phi_r({\bf S})$ has independent components. Moreover, we show that if ${\bf S}$ is a random vector in $(0,\infty)^p$ and for any leaf $r$ of the tree the components of $\Phi_r({\bf S})$ are independent, then one of these components has a Gamma distribution and the remaining $p-1$ components have Kummer distributions. Our point of departure is a relatively simple independence property due to \cite{HV15}. It states that if $X$ and $Y$ are independent random variables having Kummer and Gamma distributions (with suitably related parameters) and $T:(0,\infty)^2\to(0,\infty)^2$ is the involution defined by $T(x,y) =(y/(1+x), x+xy/(1+x))$, then the random vector $T(X,Y)$ has also independent components with Kummer and gamma distributions. By a method inspired by a proof of a similar result for the MY property, we show that this independence property characterizes the gamma and Kummer laws., Comment: 17 pages, 2 figures

Quantitative methods for studying biodiversity have been traditionally rooted in the classical theory of finite frequency tables analysis. However, with the help of modern experimental tools, like high throughput sequencing, we now begin to unlock the outstanding diversity of genomic data in plants and animals reflective of the long evolutionary history of our planet. This molecular data often defies the classical frequency/contingency tables assumptions and seems to require sparse tables with very large number of categories and highly unbalanced cell counts, e.g., following heavy tailed distributions (for instance, power laws). Motivated by the molecular diversity studies, we propose here a frequency-based framework for biodiversity analysis in the asymptotic regime where the number of categories grows with sample size (an infinite contingency table). Our approach is rooted in information theory and based on the Gaussian limit results for the effective number of species (the Hill numbers) and the empirical Renyi entropy and divergence. We argue that when applied to molecular biodiversity analysis our methods can properly account for the complicated data frequency patterns on one hand and the practical sample size limitations on the other. We illustrate this principle with two specific RNA sequencing examples: a comparative study of T-cell receptor populations and a validation of some preselected molecular hepatocellular carcinoma (HCC) markers.

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.

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., Comment: 5 pages

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.

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.

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.

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.

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.

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.

We consider a sequence of random variables $(R_n)$ defined by the recurrence $R_n=Q_n+M_nR_{n-1}$, $n\ge1$, where $R_0$ is arbitrary and $(Q_n,M_n)$, $n\ge1$, are i.i.d. copies of a two-dimensional random vector $(Q,M)$, and $(Q_n,M_n)$ is independent of $R_{n-1}$. It is well known that if $E{\ln}|M|<0$ and $E{\ln^+}|Q|<\infty$, then the sequence $(R_n)$ converges in distribution to a random variable $R$ given by $R\stackrel{d}{=}\sum_{k=1}^{\infty}Q_k\prod_{j=1}^{k-1}M_j$, and usually referred to as perpetuity. In this paper we consider a situation in which the sequence $(R_n)$ itself does not converge. We assume that $E{\ln}|M|$ exists but that it is non-negative and we ask if in this situation the sequence $(R_n)$, after suitable normalization, converges in distribution to a non-degenerate limit., Comment: Published in at http://dx.doi.org/10.3150/10-BEJ297 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

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.

If the space $\mathcal{Q}$ of quadratic forms in $\mathbb{R}^n$ is splitted in a direct sum $\mathcal{Q}_1\oplus...\oplus \mathcal{Q}_k$ and if $X$ and $Y$ are independent random variables of $\mathbb{R}^n$, assume that there exist a real number $a$ such that $E(X|X+Y)=a(X+Y)$ and real distinct numbers $b_1,...,b_k$ such that $E(q(X)|X+Y)=b_iq(X+Y)$ for any $q$ in $\mathcal{Q}_i.$ We prove that this happens only when $k=2$, when $\mathbb{R}^n$ can be structured in a Euclidean Jordan algebra and when $X$ and $Y$ have Wishart distributions corresponding to this structure., Comment: 11 pages

Free quadratic harness is a Markov process from the class of quadratic harnesses, i.e. processes with linear regressions and quadratic conditional variances. The process has recently been constructed for a restricted range of parameters in the paper "Askey-Wilson polynomials, quadratic harnesses and martingales" by W. Bryc and J. Weso{\l}owski using Askey--Wilson polynomials. Here we provide a self-contained construction of the free quadratic harness for all values of parameters.

We consider the tail behavior of random variables $R$ which are solutions of the distributional equation $R\stackrel{d}{=}Q+MR$, where $(Q,M)$ is independent of $R$ and $|M|\le 1$. Goldie and Gr\"{u}bel showed that the tails of $R$ are no heavier than exponential and that if $Q$ is bounded and $M$ resembles near 1 the uniform distribution, then the tails of $R$ are Poissonian. In this paper, we further investigate the connection between the tails of $R$ and the behavior of $M$ near 1. We focus on the special case when $Q$ is constant and $M$ is nonnegative., Comment: Published in at http://dx.doi.org/10.1214/09-AAP603 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org). This version corrects formula (6.1) in the statement of Theorem 6 in published version

We use orthogonality measures of Askey--Wilson polynomials to construct Markov processes with linear regressions and quadratic conditional variances. Askey--Wilson polynomials are orthogonal martingale polynomials for these processes., Comment: Published in at http://dx.doi.org/10.1214/09-AOP503 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

A new independence property of univariate beta distributions, related to the results of Kshirsagar and Tan for beta matrices, is presented. Conversely, a characterization of univariate beta laws through this independence property is proved. A related characterization of a family of $2\times2$ random matrices including beta matrices is also obtained. The main technical challenge was a problem involving the solution of a related functional equation., Comment: Published in at http://dx.doi.org/10.3150/07-BEJ118 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

This paper is a continuation of our previous research on quadratic harnesses, that is, processes with linear regressions and quadratic conditional variances. Our main result is a construction of a Markov process from given orthogonal and martingale polynomials. The construction uses a two-parameter extension of the Al-Salam--Chihara polynomials and a relation between these polynomials for different values of parameters., Comment: Published in at http://dx.doi.org/10.1214/009117907000000268 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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.

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.

We study a two parameter family of processes with linear regressions and linear conditional variances. We give conditions for the unique solution of this problem, and point out the connection between the resulting Markov processes and the generalized convolutions introduced by Bo\.zejko and Speicher., Comment: 12 pages, 2 figures

We show that stochastic processes with linear conditional expectations and quadratic conditional variances are Markov, and their transition probabilities are related to a three-parameter family of orthogonal polynomials which generalize the Meixner polynomials. Special cases of these processes are known to arise from the non-commutative generalizations of the Levy processes., Comment: LaTeX, 24 pages. Corrections to published version affect formulas in Theorem 4.2