We discuss a class of chain graph models for categorical variables defined by what we call a multivariate regression chain graph Markov property. First, the set of local independencies of these models is shown to be Markov equivalent to those of a chain graph model recently defined in the literature. Next we provide a parametrization based on a sequence of generalized linear models with a multivariate logistic link function that captures all independence constraints in any chain graph model of this kind.

A set of independence statements may define the independence structure of interest in a family of joint probability distributions. This structure is often captured by a graph that consists of nodes representing the random variables and of edges that couple node pairs. One important class contains regression graphs. Regression graphs are a type of so-called chain graph and describe stepwise processes, in which at each step single or joint responses are generated given the relevant explanatory variables in their past. For joint densities that result after possible marginalising or conditioning, we introduce summary graphs. These graphs reflect the independence structure implied by the generating process for the reduced set of variables and they preserve the implied independences after additional marginalising and conditioning. They can identify generating dependences that remain unchanged and alert to possibly severe distortions due to direct and indirect confounding. Operators for matrix representations of graphs are used to derive these properties of summary graphs and to translate them into special types of paths in graphs.

We consider a sequence of random variables ($R_n$) defined by the recurrence $R_n = Q_n + M_nR_{n−1}, n ≥ 1$, where $R_0$ is arbitrary and ($Q_n, M_n), n ≥ 1$, 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 $E \ln^+|Q| < ∞$, 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.

Marginal structural models were introduced in order to provide estimates of causal effects from interventions based on observational studies in epidemiological research. The key point is that this can be understood in terms of Girsanov’s change of measure. This offers a mathematical interpretation of marginal structural models that has not been available before. We consider both a model of an observational study and a model of a hypothetical randomized trial. These models correspond to different martingale measures – the observational measure and the randomized trial measure – on some underlying space. We describe situations where the randomized trial measure is absolutely continuous with respect to the observational measure. The resulting continuous-time likelihood ratio process with respect to these two probability measures corresponds to the weights in discrete-time marginal structural models. In order to do inference for the hypothetical randomized trial, we can simulate samples using observational data weighted by this likelihood ratio.

The question of existence and properties of stationary solutions to Langevin equations driven by noise processes with stationary increments is discussed, with particular focus on noise processes of pseudo-moving-average type. On account of the Wold–Karhunen decomposition theorem, such solutions are, in principle, representable as a moving average (plus a drift-like term) but the kernel in the moving average is generally not available in explicit form. A class of cases is determined where an explicit expression of the kernel can be given, and this is used to obtain information on the asymptotic behavior of the associated autocorrelation functions, both for small and large lags. Applications to Gaussian- and Lévy-driven fractional Ornstein–Uhlenbeck processes are presented. A Fubini theorem for Lévy bases is established as an element in the derivations.

Let $ξ_i, i ∈ ℕ$, be independent copies of a Lévy process {$ξ(t), t ≥ 0$}. Motivated by the results obtained previously in the context of the random energy model, we prove functional limit theorems for the process $$Z_N(t)=\sum_{i=1}^N \mathrm{e}^{\xi_i(s_N+t)}$$ as $N → ∞$, where $s_N$ is a non-negative sequence converging to $+∞$. The limiting process depends heavily on the growth rate of the sequence $s_N$. If $s_N$ grows slowly in the sense that $\lim \inf _{N→∞ } \log N / s_N > λ_2$ for some critical value $λ_2 > 0$, then the limit is an Ornstein–Uhlenbeck process. However, if $λ := \lim _{N→∞ }\log N /s_N ∈ (0, λ_2)$, then the limit is a certain completely asymmetric $α$-stable process $\mathbb{Y}_{\alpha ;\xi}$.

In this paper we study the limiting distributions of the least-squares estimators for the non-stationary first-order threshold autoregressive (TAR(1)) model. It is proved that the limiting behaviors of the TAR(1) process are very different from those of the classical unit root model and the explosive AR(1).

Let $\mathscr{P}(E)$ be the space of probability measures on a measurable space $(E,\mathcal{E})$. In this paper we introduce a class of nonlinear Markov chain Monte Carlo (MCMC) methods for simulating from a probability measure $\pi\in\mathscr{P}(E)$. Nonlinear Markov kernels (see [Feynman–Kac Formulae: Genealogical and Interacting Particle Systems with Applications (2004) Springer]) $K:\mathscr{P}(E)\times E\rightarrow\mathscr{P}(E)$ can be constructed to, in some sense, improve over MCMC methods. However, such nonlinear kernels cannot be simulated exactly, so approximations of the nonlinear kernels are constructed using auxiliary or potentially self-interacting chains. Several nonlinear kernels are presented and it is demonstrated that, under some conditions, the associated approximations exhibit a strong law of large numbers; our proof technique is via the Poisson equation and Foster–Lyapunov conditions. We investigate the performance of our approximations with some simulations.

Discrete stability extends the classical notion of stability to random elements in discrete spaces by defining a scaling operation in a randomised way: an integer is transformed into the corresponding binomial distribution. Similarly defining the scaling operation as thinning of counting measures we characterise the corresponding discrete stability property of point processes. It is shown that these processes are exactly Cox (doubly stochastic Poisson) processes with strictly stable random intensity measures. We give spectral and LePage representations for general strictly stable random measures without assuming their independent scattering. As a consequence, spectral representations are obtained for the probability generating functional and void probabilities of discrete stable processes. An alternative cluster representation for such processes is also derived using the so-called Sibuya point processes, which constitute a new family of purely random point processes. The obtained results are then applied to explore stable random elements in discrete semigroups, where the scaling is defined by means of thinning of a point process on the basis of the semigroup. Particular examples include discrete stable vectors that generalise discrete stable random variables and the family of natural numbers with the multiplication operation, where the primes form the basis.

We compare weighted sums of i.i.d. positive random variables according to the usual stochastic order. The main inequalities are derived using majorization techniques under certain log-concavity assumptions. Specifically, let $Y_i$ be i.i.d. random variables on R₊. Assuming that $\log Y_i$ has a log-concave density, we show that $∑ a_iY_i$ is stochastically smaller than ∑ b_iY_i, if ($\log a_1$, …, $\log a_n$) is majorized by ($\log b_1$, …, $\log b_n$). On the other hand, assuming that $Y_i^p$ has a log-concave density for some $p > 1$, we show that $∑a_iY_i$ is stochastically larger than $∑ b_iY_i$, if ($a_1^q$, …, $a_n^q$) is majorized by ($b_1^q$, …, $b_n^q$), where $p^{−1} + q^−1 = 1$. These unify several stochastic ordering results for specific distributions. In particular, a conjecture of Hitczenko [Sankhyā A 60 (1998) 171–175] on Weibull variables is proved. Potential applications in reliability and wireless communications are mentioned.

We show somewhat unexpectedly that whenever a general Bernstein-type maximal inequality holds for partial sums of a sequence of random variables, a maximal form of the inequality is also valid.

The objective of this paper is to provide, for the problem of univariate symmetry (with respect to specified or unspecified location), a concept of optimality, and to construct tests achieving such optimality. This requires embedding symmetry into adequate families of asymmetric (local) alternatives. We construct such families by considering non-Gaussian generalizations of classical first-order Edgeworth expansions indexed by a measure of skewness such that (i) location, scale and skewness play well-separated roles (diagonality of the corresponding information matrices) and (ii) the classical tests based on the Pearson–Fisher coefficient of skewness are optimal in the vicinity of Gaussian densities.

Multivariate versions of classical orthogonal polynomials such as Jacobi, Hahn, Laguerre and Meixner are reviewed and their connection explored by adopting a probabilistic approach. Hahn and Meixner polynomials are interpreted as posterior mixtures of Jacobi and Laguerre polynomials, respectively. By using known properties of gamma point processes and related transformations, a new infinite-dimensional version of Jacobi polynomials is constructed with respect to the size-biased version of the Poisson–Dirichlet weight measure and to the law of the gamma point process from which it is derived.