Stationary distributions of the simplest dynamical systems in R perturbed by generalized Poisson process

Let z(t) Rⁿ be a generalized Poisson process with parameter λ and let A: Rⁿ → Rⁿ be a linear operator. The conditions of existence and limiting properties as λ → ∞ or as λ → 0 of the stationary distribution of the process x(t) Rⁿ which satisfies the equation dx(t) = Ax(t)dt + dz(t) are investigated.     

Domains of attraction and regular variation in IRd

A general form for regular variation in IR² is introduced and applied to domains of attraction of stable distribution in IR² where the components have different indices. The situation in IR^d with d > 2 is more complicated but not essentially different. For simplicity this paper is limited to IR².     

Stationary probability distributions for multiresponse linear learning models

A necessary and sufficient condition is given for the existence of stationary probability distributions of a non-Markovian model with linear transition rule. Similar to the Markovian case, stationary probability distributions are characterized as eigenvectors of nonnegative matrices. The model studied includes as special cases the Markovian model as well as the linear learning model and has applications in psychological and biological research, in control theory, and in adaption theory.     

Functions of event variables of a random system with complete connections

In applications it often occurs that the experimenter is faced with functions of random processes. Suppose, for instance, that he only can draw partial or incomplete information about the underlying process or that he has to classify events for the sake of efficiency. We assume that the underlying process is a random system with complete connections (which contains the Markovian case as a special one) satisfying some basic properties, and that a mapping operates on the event space. With these two elements we construct in Section 2 a new random system with complete connections which inherits the properties of the old one (Theorem 2.2.3). In Section 3 we prove a weak convergence theorem (Theorem 3.4.4) in the theoretical framework of the so-called distance diminishing models, which gives a straightforward application in Section 4 to conditional probabilities related to partially observed events (Theorems 4.1.3). Finally we prove a Shannon-McMillan-type theorem (Theorem 4.2.3) finding application to classification procedures.     

Stationary isothermic surfaces and some characterizations of the hyperplane in the N-dimensional Euclidean space

We consider the entire graph S of a continuous real function over RN−1 with N?3. Let Ω be a domain in RN with S as a boundary. Consider in Ω the heat flow with initial temperature 0 and boundary temperature 1. The problem we consider is to characterize S in such a way that there exists a stationary isothermic surface in Ω. We show that S must be a hyperplane under some general conditions on S. This is related to Liouville or Bernstein-type theorems for some elliptic Monge-Ampère-type equation.     

Conditioned rates of convergence in the CLT for sums and maximum sums

An interesting recent result of Landers and Roggé (1977, Ann. Probability5, 595–600) is investigated further. Rates of convergence in the conditioned central limit theorem are developed for partial sums and maximum partial sums, with positive mean and zero mean separately, of sequences of independent identically distributed random variables. As corollaries we obtain a conditioned central limit theorem for maximum partial sums both for positive and zero mean cases.     

A canonical decomposition of the probability measure of sets of isotropic random points in Rn

The probability measure of X = (x₀,…, x_r), where x₀,…, x_r are independent isotropic random points in $\text{R}$ⁿ (1 ≤ r ≤ n ? 1) with absolutely continuous distributions is, for a certain class of distributions of X, expressed as a product measure involving as factors the joint probability measure of (ω, ?), the probability measure of p, and the probability measure of $\text{Y}{}^1\text{= (y}{}_0{}^1\text{,…, y}{}_{\mathrm{r}}{}^1\text{)}$. Here ω is the r-subspace parallel to the r-flat η determined by X, ? is a unit vector in ω^⊥ with ‘initial’ point at the origin [ω^⊥ is the (n ? r)-subspace orthocomplementary to ω], p is the norm of the vector z from the origin to the orthogonal projection of the origin on η, and $\text{y}{}_{\mathrm{i}}{}^1\text{=}\text{(x}{}_{\mathrm{i}}\text{? z)}\text{α(p}{}^2\text{)}$, where α is a scale factor determined by p. The probability measure for ω is the unique probability measure on the Grassmann manifold of r-subspaces in $\text{R}$ⁿ invariant under the group of rotations in $\text{R}$ⁿ, while the conditional probability measure of ? given ω is uniform on the boundary of the unit (n ? r)-ball in ω^⊥ with centre at the origin. The decomposition allows the evaluation of the moments, for a suitable class of distributions of X, of the r-volume of the simplicial convex hull of {x₀,…, x_r} for 1 ≤ r ≤ n.     

Analysis of random walks in dynamic random environments via L2-perturbations

We consider random walks in dynamic random environments given by Markovian dynamics on ${\mathbb{Z}}^d$. We assume that the environment has a stationary distribution $\mu$ and satisfies the Poincaré inequality w.r.t. $\mu$. The random walk is a perturbation of another random walk (called “unperturbed”). We assume that also the environment viewed from the unperturbed random walk has stationary distribution $\mu$. Both perturbed and unperturbed random walks can depend heavily on the environment and are not assumed to be finite-range. We derive a law of large numbers, an averaged invariance principle for the position of the walker and a series expansion for the asymptotic speed. We also provide a condition for non-degeneracy of the diffusion, and describe in some details equilibrium and convergence properties of the environment seen by the walker. All these results are based on a more general perturbative analysis of operators that we derive in the context of $L^2$- bounded perturbations of Markov processes by means of the so-called Dyson–Phillips expansion.     

Linear instability of relative equilibria for n-body problems in the plane

Following Smale, we study simple symmetric mechanical systems of n point particles in the plane. In particular, we address the question of the linear and spectral stability properties of relative equilibria, which are special solutions of the equations of motion.