Strong consistency of least squares estimates in multiple regression II

The strong consistency of least squares estimates in multiple regression models is established under minimal assumptions on the design and weak dependence and moment restrictions on the errors.     

Diffusion approximation for storage processes

For a sequence of storage processes with general release rate functions which contain, as a special case, queuing processes, we show that under appropriate conditions, suitably normalized processes for storage processes converge to diffusions in the sense of law.     

Estimation of vector ARMAX models

The asymptotic properties of maximum likelihood estimates of a vector ARMAX system are considered under general conditions, relating to the nature of the exogenous variables and the innovation sequence and to the form of the parameterization of the rational transfer functions, from exogenous variables and innovations to the output vector. The exogenous variables are assumed to be such that the sample serial covariances converge to limits. The innovations are assumed to be martingale differences and to be nondeterministic in a fairly weak sense. Stronger conditions ensure that the asymptotic distribution of the estimates has the same covariance matrix as for Gaussian innovations but these stronger conditions are somewhat implausible. With each ARMAX structure may be associated an integer (the McMillan degree) and all structures for a given value of this integer may be topologised as an analytic manifold. Other parameterizations and topologisations of spaces of structures as analytic manifolds may also be considered and the presentation is sufficiently general to cover a wide range of these. Greater generality is also achieved by allowing for general forms of constraints.     

Central limit theorem and weak law of large numbers with rates for martingales in Banach spaces

This paper is concerned with large-$\text{O}$ error estimates concerning convergence in distribution as well as norm convergence for Banach space-valued martingale difference sequences. Indeed, two general limit theorems equipped with rates of convergence for such difference sequences are established. Applications of these lead to the central limit theorem and the weak law of large numbers with rates for Banach space-valued martingales.     

On the structure of random plane-oriented recursive trees and their branches

This paper is an investigation of the structural properties of random plane-oriented recursive trees and their branches. We begin by an enumeration of these trees and some general properties related to the outdegrees of nodes. Using generalized Pólya urn models we study the exact and limiting distributions of the size and the number of leaves in the branches of the tree. The exact distribution for the leaves in the branches is given by formulas involving second-order Eulerian numbers. A martingale central limit theorem for a linear combination of the number of leaves and the number of internal nodes is derived. The distribution of that linear combination is a mixture of normals with a beta distribution as its mixing density. The martingale central limit theorem allows easy determination of the limit laws governing the leaves in the branches. Furthermore, the asymptotic joint distribution of the number of nodes of outdegree 0, 1 and 2 is shown to be trivariate normal. © 1993 John Wiley & Sons, Inc.     

The maximum of a Lévy process reflected at a general barrier

We investigate the reflection of a Lévy process at a deterministic, time-dependent barrier and in particular properties of the global maximum of the reflected Lévy process. Under the assumption of a finite Laplace exponent, ψ(θ)$\psi \left(\theta \right)$, and the existence of a solution θ^∗>0${\theta }^{\ast }>0$ to ψ(θ)=0$\psi \left(\theta \right)=0$ we derive conditions in terms of the barrier for almost sure finiteness of the maximum. If the maximum is finite almost surely, we show that the tail of its distribution decays like Kexp(−θ^∗x)$Kexp(-{\theta }^{\ast }x)$. The constant K$K$ can be completely characterized, and we present several possible representations. Some special cases where the constant can be computed explicitly are treated in greater detail, for instance Brownian motion with a linear or a piecewise linear barrier. In the context of queuing and storage models the barrier has an interpretation as a time-dependent maximal capacity. In risk theory the barrier can be interpreted as a time-dependent strategy for (continuous) dividend pay out.     

A Banach lattice approach to convergent integrably bounded set-valued martingales and their positive parts

Denote by cf(X) the set of all nonempty convex closed subsets of a separable Banach space X. Let (Ω,Σ,μ) be a complete probability space and denote by (L¹[Σ,cf(X)],Δ) the complete metric space of (equivalence classes of a.e. equal) integrably bounded cf(X)-valued functions. For any preassigned filtration (Σi), we describe the space of Δ-convergent integrably bounded cf(X)-valued martingales in terms of the Δ-closure of in L¹[Σ,cf(X)]. In particular, we provide a formula to calculate the join of two such martingales and the positive part of such a martingale. Our object is achieved by considering the more general setting of a near vector lattice (S,d), endowed with a Riesz metric d. By means of Rådström's embedding theorem for such spaces, a link is established between the space of convergent martingales in S and the space of convergent martingales in the Rådström completion R(S) of S. This link provides information about the former space of martingales, via known properties of measure-free martingales in Riesz normed vector lattices, applicable to R(S). We also apply our general results to the spaces of Δ-convergent ck(X)-valued martingales, where ck(X) denotes the set of all nonempty convex compact subsets of X.     

Nonparametric estimation for censored lifetimes suffering from unknown selection bias

In a population of individuals, where the random variable (r.v.) σ denotes the birth time and X the lifetime, we consider the case, where an individual can be observed only if its life-line (σ, X) = {(σ + y, y), 0 ≤ y ≤ X} intersects a given Borel set S in ℝ × ℝ₊. Denoting by σ _S and X _S the birth time and lifetime for the observed individuals, we point out that the distribution function (d.f.) F _S of the r.v. X _S suffers from a selection bias in the sense that F _S = ∝ w d F/μ _S, where w and μ _S depend only on the distribution of σ and on F, the d.f. of X. Assuming in addition that the r.v. X _S is randomly right-censored as soon as the individual is selected, we construct a productlimit estimator for the d.f. F _S and a nonparametric estimator ŵ for the weight function w. We prove a consistency result for ŵ and a weak convergence result for . We establish in addition an exponential bound for .      

一类随机适应序列部分和的局部极限定理的一个注记

研究了一类随机适应序列的强极限定理,推广了最近发表的几个结果,并进一步推广了Borel—Cantelli引理．     

Probabilistic gradient approximation for a viscous scalar conservation law in space dimension d≥2

In this paper, we are interested in the solution of a viscous scalar conservation law. We remark that its first order spatial derivatives solve a system of partial differential equations presenting a nonlocal nonlinearity. We associate a nonlinear martingale problem with this system. After proving existence and uniqueness for the martingale problem, we obtain a propagation of chaos result for a system of interacting diffusion processes. We deduce that it is possible to approximate the solution of the viscous scalar conservation law thanks to the interacting diffusions