Local empirical processes near boundaries of convex bodies

We investigate the behaviour of Poisson point processes in the neighbourhood of the boundary ∂K of a convex body K in ,d ≥ 2. Making use of the geometry of K, we show various limit results as the intensity of the Poisson process increases and the neighbourhood shrinks to ∂K. As we shall see, the limit processes live on a cylinder generated by the normal bundle of K and have intensity measures expressed in terms of the support measures of K. We apply our limit results to a spatial version of the classical change-point problem, in which random point patterns are considered which have different distributions inside and outside a fixed, but unknown convex body K.     

Self-Intersection Local Time for Some S′(ℝd)-Ornstein-Uhlenbeck Processes Related to Inhomogeneous Fields

We study existence and path continuity of the self-intersection local time (SILT) for some S′(ℝ^d)-valued Ornstein-Uhlenbeck processes. Examples of such processes arise in particular as fluctuation limits ofparticle systems. We analyze the effect that different types ofmeasures involved in the covariances of the processes have on existence and continuity of SILT. These measures do not necessarily have the regularity and homogeneity properties of those that have been considered before; this precludes some key ingredients of the previous techniques. We develop new technical tools and prove sufficient conditions and some necessary conditions for existence of SILT, and sufficient conditions for path continuity of SILT. The questions of existence and continuity involve problems of existence of integrals and singular integrals. We give examples which illustrate how different types of measures e.g., atomic or with L²-density) may produce different critical dimensions for existence of SILT. One of our motivations is the desire for a better understanding ofwhat SILT represents for S′(ℝ^d)-valued processes.     

Semi-Selfsimilar Processes

A notion of semi-selfsimilarity of R ^d -valued stochastic processes is introduced as a natural extension of the selfsimilarity. Several topics on semi-selfsimilar processes are studied: the existence of the exponent for semi-selfsimilar processes; characterization of semi-selfsimilar processes as scaling limits; relationship between semi-selfsimilar processes with independent increments and semi-selfdecomposable distributions, and examples; construction of semi-selfsimilar processes with stationary increments; and extension of the Lamperti transformation. Semi-stable processes where all joint distributions are multivariate semi-stable are also discussed in connection with semi-selfsimilar processes. A wide-sense semi-selfsimilarity is defined and shown to be reducible to semi-selfsimilarity.     

Limit Processes for Age-Dependent Branching Particle Systems

We consider systems of spatially distributed branching particles in R ^d . The particle lifelengths are of general form, hence the time propagation of the system is typically not Markov. A natural time-space-mass scaling is applied to a sequence of particle systems and we derive limit results for the corresponding sequence of measure-valued processes. The limit is identified as the projection on R ^d of a superprocess in R ₊×R ^d . The additive functional characterizing the superprocess is the scaling limit of certain point processes, which count generations along a line of descent for the branching particles.     

Decay parameter and related properties of 2-type branching processes

We consider the decay parameter, invariant measures/vectors and quasi-stationary dis- tributions for 2-type Markov branching processes. Investigating such properties is crucial in realizing life period of branching models. In this paper, some important properties of the generating functions for 2-type Markov branching q-matrix are firstly investigated in detail. The exact value of the decay parameter λC of such model is given for the communicating class C = Z+2 \ 0. It is shown that this λC can be directly ...     

Functional limit theorems for the “disorder” problem

Let X ⁽ⁿ⁾=(X _k ), 1≦k≦n be random process with discrete time defined by its transition probabilities which belong to some parametric family. It is assumed that the parameters of the transition probabilities before and/or after disorder as well as the disorder time, are unknown. For statistical purposes the processes of Radon-Nikodym derivatives of the measures generated by processes with disorder at the time s with respect to the measure generated by process without disorder where 1≦s≦n are often used. In the paper general sufficient conditions are given for weak convergence of these processes. Some examples are given to illustrate the application of the results obtained.     

Branching random walks and contact processes on homogeneous trees

Summary. Branching random walks and contact processes on the homogeneous tree in which each site has d+1 neighbors have three possible types of behavior (for d≧ 2): local survival, local extinction with global survival, and global extinction. For branching random walks, we show that if there is local extinction, then the probability that an individual ever has a descendent at a site n units away from that individual’s location is at most d ^{− n/2} , while if there is global extinction, this probability is at most d ⁻ⁿ . Next, we consider the structure of the set of invariant measures with finite intensity for the system, and see how this structure depends on whether or not there is local and/or global survival. These results suggest some problems and conjectures for contact processes on trees. We prove some and leave others open. In particular, we prove that for some values of the infection parameter λ, there are nontrivial invariant measures which have a density tending to zero in all directions, and hence are different from those constructed by Durrett and Schinazi in a recent paper. Received: 26 April 1996/In revised form: 20 June 1996     

On bulk-service MAP/PH L,N /1/N G-Queues with repeated attempts

We consider a retrial queue with a finite buffer of size N, with arrivals of ordinary units and of negative units (which cancel one ordinary unit), both assumed to be Markovian arrival processes. The service requirements are of phase type. In addition, a PH^L,N bulk service discipline is assumed. This means that the units are served in groups of size at least L, where 1≤ L≤ N. If at the completion of a service fewer than L units are present at the buffer, the server switches off and waits until the buffer length reaches the threshold L. Then it switches on and initiates service for such a group of units. On the contrary, if at the completion of a service L or more units are present at the buffer, all units enter service as a group. Units arriving when the buffer is full are not lost, but they join a group of unsatisfied units called “orbit”. Our interest is in the continuous-time Markov chain describing the state of the queue at arbitrary times, which constitutes a level dependent quasi-birth-and-death process. We start by analyzing a simplified version of our queueing model, which is amenable to numerical calculation and is based on spatially homogeneous quasi-birth-and-death processes. This leads to modified matrix-geometric formulas that reveal the basic qualitative properties of our algorithmic approach for computing performance measures. AMS Subject Classification: Primary 60K25 Secondary 68M20 90B22.     

Hyperuniform and rigid stable matchings

We study a stable partial matching τ of the d‐dimensional lattice with a stationary determinantal point process Ψ on R^d with intensity α>1. For instance, Ψ might be a Poisson process. The matched points from Ψ form a stationary and ergodic (under lattice shifts) point process Ψ^τ with intensity 1 that very much resembles Ψ for α close to 1. On the other hand Ψ^τ is hyperuniform and number rigid, quite in contrast to a Poisson process. We deduce these properties by proving more general results for a stationary point process Ψ, whose so‐called matching flower (a stopping set determining the matching partner of a lattice point) has a certain subexponential tail behavior. For hyperuniformity, we also additionally need to assume some mixing condition on Ψ. Furthermore, if Ψ is a Poisson process then Ψ^τ has an exponentially decreasing truncated pair correlation function.     

Inference for thinned point processes, with application to Cox processes

Given i.i.d. point processes N₁, N₂,…, let the observations be p-thinnings N′₁, N′₂,…, where p is a function from the underlying space E (a compact metric space) to [0, 1], whose interpretation is that a point of N_i at x is retained with probability p(x) and deleted with probability 1−p(x). Strongly consistent estimators of the thinning function p and the Laplace functional L_N(f) = E[e^−N(f)] of the N_i are constructed; associated “central limit” properties are given. Tests are presented, for the case when the N_i and N′_i are both observable, of the hypothesis that the N′_i are p-thinnings of the N_i. State estimation techniques are developed for the case where the N_i are Cox processes directed by unobservable random measures M_i; these techniques yield minimum mean-squared error estimators, based on observation of only the thinned processes N′_i of the N_i and the directing measures M_i. Limit theorems for empirical Laplace functionals of point processes are given.     

On stochastic processes associated with relativistic stable distributions

Lévy processes with marginal relativistic α-stable distributions are described. Strictly stationary Ornstein-Uhlenbeck type processes with one-dimentional relativistic α-stable distributions are constructed. The exponential family as Esscher transforms of distributions on D _[0,∞)(R ^d ) of relativistic α-stable Lévy processes is obtained and the corresponding mixed exponential processes are characterized.     

The Exact Hausdorff Measure Function of the Level Sets of Multi-parameter Symmetric Stable Process

In this paper, we investigate the Hausdorff measure for level sets of N-parameter Rd-valued stable processes, and develop a means of seeking the exact Hausdorff measure function for level sets of N-parameter Rd-valued stable processes. We show that the exact Hausdorff measure function of level sets of N-parameter Rd-valued symmetric stable processes of index α is Ф（r） = r^N-d/α （log log l/r）d/α when Nα 〉 d. In addition, we obtain a sharp lower bound for the Hausdorff measure of level sets of general （N, d, α） strictly stable processes.     

Hamilton-Jacobi-Bellman inequality for the average control of piecewise deterministic Markov processes

ABSTRACT

The main goal of this paper is to study the infinite-horizon long run average continuous-time optimal control problem of piecewise deterministic Markov processes (PDMPs) with the control acting continuously on the jump intensity λ and on the transition measure Q of the process. We provide conditions for the existence of a solution to an integro-differential optimality inequality, the so called Hamilton-Jacobi-Bellman (HJB) equation, and for the existence of a deterministic stationary optimal policy. These results are obtained by using the so-called vanishing discount approach, under some continuity and compactness assumptions on the parameters of the problem, as well as some non-explosive conditions for the process.     

Fractional Brownian Density Process and Its Self-Intersection Local Time of Order k

The fractional Brownian density process is a continuous centered Gaussian ( ^d )-valued process which arises as a high-density fluctuation limit of a Poisson system of independent d-dimensional fractional Brownian motions with Hurst parameter H. ( ( ^d ) is the space of tempered distributions). The main result proved in the paper is that if the intensity measure of the (initial) Poisson random measure on ^d is either the Lebesgue measure or a finite measure, then the density process has self-intersection local time of order k 2 if and only if Hd < k/(k – 1). The latter is also the necessary and sufficient condition for existence of multiple points of order k for d-dimensional fractional Brownian motion, as proved by Talagrand¹². This result extends to a non-Markovian case the relationship known for (Markovian) symmetric -stable Lévy processes and their corresponding density processes. New methods are used in order to overcome the lack of Markov property. Other properties of the fractional Brownian density process are also given, in particular the non-semimartingale property in the case H 1/2, which is obtained by a general criterion for the non-semimartingale property of real Gaussian processes that we also prove.     

Nonparametric Estimation of Regression Functions in Point Process Models

We prove that the empirical L ²-risk minimizing estimator over some general type of sieve classes is universally, strongly consistent for the regression function in a class of point process models of Poissonian type (random sampling processes). The universal consistency result needs weak assumptions on the underlying distributions and regression functions. It applies in particular to neural net classes and to radial basis function nets. For the estimation of the intensity functions of a Poisson process a similar technique yields consistency of the sieved maximum likelihood estimator for some general sieve classes. This revised version was published online in August 2006 with corrections to the Cover Date.     

Path properties of an infinite system of Wiener processes

Let be a Poisson process on ^d of intensity and letW ₁(t),W ₂ (t),..., be a sequence of independent Wiener processes. LetW _i (t)=X _i +W _i (t) whereX ₁,X ₂,..., are the points of . Consider the processess(t)=#{i:X _i (t)1}. These and related processes are studied.     

Symmetrizations of Markov processes

Two methods for symmetrizing Markov processes are discussed. Letu ^a(x, y) be the potential density of a Lévy process on a compact Abelian groupG. A general condition is given that guarantees thatv(x, y)=u^a(x, y)+u^a(y, x) is the potential density of a symmetric Lévy process onG. The second method arises by considering the linear space of one-potentialsU ¹ f, withf inL ², endowed with the inner product (U ¹ f,U ¹ g)=fU ¹ g+gU ¹ f. If the semigroup ofX(t) is normal, then the completionH of this space is the Dirichlet space of a symmetric processY(t). A set that is semipolar forX(t) is polar forY(t).     

Discontinuous Measure-Valued Branching Processes and Generalized Stochastic Equations

We study a class of integrable and discontinuous measure-valued branching processes. They are constructed as limits of renormalized spatial branching processes, the underlying branching distribution belonging to the domain of attraction of a stable law. These processes, computed on a test function f, are semimartingales whose martingale terms are identified with integrals of f with respect to a martingale measure. According to a representation theorem of continuous (respectively purely discontinuous) martingale measures as stochastic integrals with respect to a white noise (resp. to a POISSON process), we prove that the measure-valued processes that we consider are solutions of stochastic differential equations in the space of L² (Ω)-valued vector measures.     

On Some Inequalities for Poisson Networks

A stationary Poisson hyperplane process in R^d induces a random network of (d-2)-flats, each of which is the intersection of two hyperplanes of the process. It is known that the intensity of the induced (d-2)-flat process divided by the square of the intensity of the original hyperplane process is maximal in the isotropic case. An integral-geometric formula for elliptic spaces is presented, from which the mentioned extremum property and related inequalities for superpositions of stationary Poisson hyperplane processes are derived.