Tail homogeneity of invariant measures of multidimensional stochastic recursions in a critical case

We consider the stochastic recursion ${X_{n+1} = M_{n+1}X_{n} + Q_{n+1}, (n \in \mathbb{N})}$ , where ${Q_n, X_n \in \mathbb{R}^d }$ , M _n are similarities of the Euclidean space ${ \mathbb{R}^d }$ and (Q _n , M _n ) are i.i.d. We study asymptotic properties at infinity of the invariant measure for the Markov chain X _n under assumption ${\mathbb{E}{[\log|M|]}=0}$ i.e. in the so called critical case.     

Convergence to stable laws for a class of multidimensional stochastic recursions

We consider a Markov chain ${\{X_n\}_{n=0}^\infty}$ on ${\mathbb R^d}$ defined by the stochastic recursion X _n  = M _n X _n-1 + Q _n , where (Q _n , M _n ) are i.i.d. random variables taking values in the affine group ${A(\mathbb R^d)=\mathbb R^d\rtimes {\rm GL}(\mathbb R^d)}$ . Assume that M _n takes values in the group of similarities of ${\mathbb R^d}$ , and the Markov chain has a unique stationary measure ν, which has unbounded support. We denote by |M _n | the expansion coefficient of M _n and we assume ${\mathbb E [|M|^\alpha]=1}$ for some positive α. We show that the partial sums ${S_n=\sum_{k=0}^n X_k}$ , properly normalized, converge to a normal law (α ≥ 2) or to an infinitely divisible law, which is stable in a natural sense (α < 2). These laws are fully nondegenerate, if ν is not supported on an affine hyperplane. Under an aperiodicity hypothesis, we prove also a local limit theorem for the sums S _n . If α ≤ 2, proofs are based on the homogeneity at infinity of ν and on a detailed spectral analysis of a family of Fourier operators P _v considered as perturbations of the transition operator P of the chain {X _n }. The characteristic function of the limit law has a simple expression in terms of moments of ν (α > 2) or of the tails of ν and of stationary measure for an associated Markov operator (α ≤ 2). We extend the results to the situation where M _n is a random generalized similarity.     

On stochastic boundedness and stationary measures for Markov processes

In this report we relate the property of stochastic boundedness to the existence of stationary measures for arbitrary Markov processes on the positive real line. We further develop a sufficiency criterion for the independence of such measures from initial conditions. The results are then applied to the question of approximating the fixed point vector of an irreducible infinite stochastic matrix by the solutions of finite ones.     

Improved recursions for some compound poisson distributions

Calculations by means of the well-known recursion for the compound Poisson distribution are in general time-consuming since each probability depends on all the preceding ones. It is shown that for some claim size distributions a more efficient recursion can be derived.     

On limiting behavior of stationary measures for stochastic evolution systems with small noise intensity

The limiting behavior of stochastic evolution processes with small noise intensity ε is investigated in distribution-based approaches.Let μ~ε be a stationary measure for stochastic process X~ε with small ε and X~0 be a semiflow on a Polish space.Assume that {μ~ε:0 ε≤ε_0} is tight.Then all their limits in the weak sense are X~0-invariant and their supports are contained in the Birkhoff center of X~0.Applications are made to various stochastic evolution systems,including stochastic ordinary differential equations,stochastic partial differential equations,and stochastic functional differential equations driven by Brownian motion or Levy processes.     

Alternative models for stationary stochastic processes

We consider some parametric spectral estimators that can be used in a wide range of situations. Assuming the existence of fourth moments, we establish rates of convergence of the estimators, and a central limit theorem.     

Dynamic programming for multidimensional stochastic control problems

In this paper we study a general multidimensional diffusion-type stochastic control problem. Our model contains the usual regular control problem, singular control problem and impulse control problem as special cases. Using a unified treatment of dynamic programming, we show that the value function of the problem is a viscosity solution of certain Hamilton-Jacobi-Bellman (HJB) quasivariational inequality. The uniqueness of such a quasi-variational inequality is proved. Supported in part by USA Office of Naval Research grant #N00014-96-1-0262. Supported in part by the NSFC Grant #79790130, the National Distinguished Youth Science Foundation of China Grant #19725106 and the Chinese Education Ministry Science Foundation.     

Extension of stationary stochastic processes

Summary LetX be an arbitrary Hausdorff space, and consider a stationary stochastic process inX with time interval [0, 1], i.e. a tight probability onX ^{[0, 1]}, equipped with the Borel -field of the product space. We prove the existence of a stationary extension of this process to ₀ ⁺ . Furthermore, we show that the extended process may be chosen to have continuous paths if the original process has this property. Under stronger topological assumptions, we derive the corresponding results whenX ^{[0, 1]} is equipped with the product of the Borel -fields.Corporate Research and Development, SIEMENS AG, D-81730 Munich, Germany     

Inverse theorems for some multidimensional operators

In this paper we introduce a new K-functional which is especially useful for non-Feller operators. Inverse theorems for multidimensional Bernstein-Sikkema and Bernstein-Durrmeyer operators are given on a simplex and on a cube. Supported by National Science Foundation of China.     

Non-uniqueness of stationary measures for self-stabilizing processes

We investigate the existence of invariant measures for self-stabilizing diffusions. These stochastic processes represent roughly the behavior of some Brownian particle moving in a double-well landscape and attracted by its own law. This specific self-interaction leads to nonlinear stochastic differential equations and permits pointing out singular phenomena like non-uniqueness of associated stationary measures. The existence of several invariant measures is essentially based on the non-convex environment and requires generalized Laplace’s method approximations.     

Martingale and stationary solutions for stochastic Navier-Stokes equations

Summary We prove the existence of martingale solutions and of stationary solutions of stochastic Navier-Stokes equations under very general hypotheses on the diffusion term. The stationary martingale solutions yield the existence of invariant measures, when the transition semigroup is well defined. The results are obtained by a new method of compactness.     

Stability of stationary rotations of multidimensional rigid body

It is well known that rotations of a free three-dimensional rigid body around the long and short axes of inertia are stable, while the rotation around the intermediate axis is unstable. We generalize this result to the case of a rigid body in a space of arbitrary dimension.     

Absolutely continuous invariant measures for multidimensional expanding maps

We investigate the existence and statistical properties of absolutely continuous invariant measures for multidimensional expanding maps with singularities. The key point is the establishment of a spectral gap in the spectrum of the transfer operator. Our assumptions appear quite naturally for maps with singularities. We allow maps that are discontinuous on some extremely wild sets, the shape of the discontinuities being completely ignored with our approach.     

Microscopic stationary states for stochastic systems with particle flux

Summary Let _t be the symmetric nearest neighbor simple exclusion process on ³ modified to allow the spontaneous creation and destruction of particles at the origin. We study the invariant measures for the system in this non-equilibrium situation, for which a net current of particles flows from the origin. We analyze the n-point correlations of the invariant measures, which for n=2 behave like max (x ₁ x ₂)^–4.Partially supported by NSF Grants: PHY 8201708, DMR 81-14726-02, and DMS 85-12505Partially supported by CNPQ GRANT No. 201682-83     

Fuzzy measures as extensions of stochastic measures

Links between fuzzy measures (cf. Höhle, Z. Wahrsch. Verw. Gebiete36 (1976), 179–188) and stochastic measures (cf. Morando, Lecture Notes in Mathematics No. 88, pp. 190–216, Springer-Verlag, Berlin/New York, 1969) are specified. In particular, a class of stochastic measures, from which fuzzy measures can be derived quite naturally, is exhibited.     

Order preservation for multidimensional stochastic functional differential equations with jumps

Sufficient and necessary conditions are presented for the order preservation of stochastic functional differential equations on ${\mathbb{R}^d}$ with non-Lipschitzian coefficients driven by the Brownian motion and Poisson processes. The sufficiency of the conditions extends and improves some known comparison theorems derived recently for one-dimensional equations and multidimensional equations without delay, and the necessity is new even in these special situations.     

Nonparametric Bayesian drift estimation for multidimensional stochastic differential equations*

We consider nonparametric Bayesian estimation of the drift coefficient of a multidimensional stochastic differential equation from discrete-time observations on the solution of this equation. Under suitable regularity conditions, we establish posterior consistency in this context.