Asymptotic Palm likelihood theory for stationary point processes

In the present paper, we propose a Palm likelihood approach as a general estimating principle for stationary point processes in $\mathbf{R}^d$ for which the density of the second-order factorial moment measure is available in closed form or in an integral representation. Examples of such point processes include the Neyman–Scott processes and the log Gaussian Cox processes. The computations involved in determining the Palm likelihood estimator are simple. Conditions are provided under which the Palm likelihood estimator is strongly consistent and asymptotically normally distributed.     

Long range percolation in stationary point processes

We consider a continuum percolation model in ?d, d ? 1 in which any two points of a stationary point process are connected with a probability which decays exponentially in the distance between the points. We give sufficient conditions for the (non)-existence of a phase transition. We also give examples of processes which show that it is impossible to write down a theorem which relates the critical parameter value of a process to its density. Finally, we show that uniqueness of the infinite cluster is still valid in this general setting. © 1993 John Wiley & Sons, Inc.     

Estimation of Palm measures of stationary point processes

Summary Estimators of the Palm measure of a stationary point process on a finite-dimensional Euclidean space are developed and shown to be strongly uniformly consistent. From them, similarly consistent estimators of reduced moment measures, the spectral measure, the spectral density function and the underlying probability measure itself are derived. Normal and Poisson approximations to distributions of estimators are presented. Application is made to the problem of combined inference and linear state estimation.Research supported by Air Force Office of Scientific Research, AFSC, grant 82-0029C. The United States Government is authorized to reproduce and distribute reprints for governmental purposes     

Absolute regularity and Brillinger-mixing of stationary point processes

We study the following problem: How to verify Brillinger-mixing of stationary point processes in $ {{\mathbb{R}}^d} $ by imposing conditions on a suitable mixing coefficient? For this, we define an absolute regularity (or β-mixing) coefficient for point processes and derive, in terms of this coefficient, an explicit condition that implies finite total variation of the kth-order reduced factorial cumulant measure of the point process for fixed $ k\geqslant 2 $ . To prove this, we introduce higher-order covariance measures and use Statulevi?ius’ representation formula for mixed cumulants in case of random (counting) measures. To illustrate our results, we consider some Brillinger-mixing point processes occurring in stochastic geometry.     

Bounds for the variance of certain stationary point processes

For the variance of stationary renewal and alternating renewal processes N_n(·) the paper establishes upper and lower bounds of the form

$\text{?B}{}_1\text{?varN}{}_8\text{(0,x–Aλx?B}{}_2\text{(0<x<∞)}$

, where λ=EN₈(0,1), with constants A, B₁ and B₂ that depend on the first three moments of the interval distributions for the processes concerned. These results are consistent with the value of the constant A for a general stationary point process suggested by Cox in 1963 [1].     

A central limit theorem for mixing stationary point processes

Suppose A₀ is a strictly stationary, second order point process on Z^d that is ?-mixing. The particles initially present are then continually subjected to random translations via random walks. If A_n is the point process resulting at time n, then we prove, under certain technical conditions, that the total occupation time by time n of a finite nonempty subset B of Z^d, namely, S_n(B)=Σⁿ_k=1A_k(B), is asymptotically normally distributed.     

Extreme value theory for continuous parameter stationary processes

Summary In this paper the central distributional results of classical extreme value theory are obtained, under appropriate dependence restrictions, for maxima of continuous parameter stochastic processes. In particular we prove the basic result (here called Gnedenko's Theorem) concerning the existence of just three types of non-degenerate limiting distributions in such cases, and give necessary and sufficient conditions for each to apply. The development relies, in part, on the corresponding known theory for stationary sequences.The general theory given does not require finiteness of the number of upcrossings of any levelx. However when the number per unit time is a.s. finite and has a finite mean(x), it is found that the classical criteria for domains of attraction apply when(x) is used in lieu of the tail of the marginal distribution function. The theory is specialized to this case and applied to give the general known results for stationary normal processes for which(x) may or may not be finite).A general Poisson convergence theorem is given for high level upcrossings, together with its implications for the asymptotic distributions ofr ^th largest local maxima.This work was supported by the Office of Naval Research under Contract N00014-75-C-0809, and in part by the Danish natural Science research Council     

On the Bayesian estimation for the stationary Neyman-Scott point processes

The pure and modified Bayesian methods are applied to the estimation of parameters of the Neyman-Scott point process. Their performance is compared to the fast, simulation-free methods via extensive simulation study. Our modified Bayesian method is found to be on average 2.8 times more accurate than the fast methods in the relative mean square errors of the point estimates, where the average is taken over all studied cases. The pure Bayesian method is found to be approximately as good as the fast methods. These methods are computationally affordable today.     

The application of the nonsmooth critical point theory to the stationary electrorheological fluids

In this paper, we prove the existence of variational solutions to systems modeling electrorheological fluids in the stationary case. Our method of proof is based on the nonsmooth critical point theory for locally Lipschitz functional and the properties of the generalized Lebesgue–Sobolev space.     

Multiplicity theory for multivariate wide sense stationary generalized processes

It is shown that the representation theory of a multivariate, purely nondeterministic, wide sense stationary generalized process can be reduced to a study of some isomorphism results established for commutation relations occurring in quantum mechanics. Using this simplification a multiplicity theory is developed. The time domain and spectral representation of the process are investigated in this context, and the concept of a generalized innovations process is introduced.     

Central limit theorems for empirical product densities of stationary point processes

We prove the asymptotic normality of kernel estimators of second- and higher-order product densities (and of the pair correlation function) for spatially homogeneous (and isotropic) point processes observed on a sampling window \(W_n\) , which is assumed to expand unboundedly in all directions as \(n \rightarrow \infty \,\) . We first study the asymptotic behavior of the covariances of the empirical product densities under minimal moment and weak dependence assumptions. The proof of the main results is based on the Brillinger-mixing property of the underlying point process and certain smoothness conditions on the higher-order reduced cumulant measures. Finally, the obtained limit theorems enable us to construct \(\chi ^2\) -goodness-of-fit tests for hypothetical product densities.     

On the entropy rate of stationary point processes and its discrete approximation

Sans résumé     

Imbedded construction of stationary sequences and point processes with a random memory

We show convergence in variation to a unique stationary state for a class of point processes (respectively, stochastic sequences) with stochastic intensity kernels (respectively, transition probabilities) including the (A,m)-processes of Lindvall [12]. This is done under two basic conditions: first, the random memory of the processes considered is consistent or non-reusable (that is, past information not used at a given time cannot be recalled at a later time) and secondly, the kernels have a deterministic fixed component for which the memory is almost surely finite.     

Equivalence of functional limit theorems for stationary point processes and their Palm distributions

Summary Let P be the distribution of a stationary point process on the real line and let P ⁰ be its Palm distribution. In this paper we consider two types of functional limit theorems, those in terms of the number of points of the point process in (0, t] and those in terms of the location of the nth point right of the origin. The former are most easily obtained under P and the latter under P ⁰. General conditions are presented that guarantee equivalence of either type of functional limit theorem under both probability measures, and under a third, P ₁, which plays a role in the proofs and is obtained from P by shifting the origin to the first point of the process on the right.In a brief final section the obtained results for either type of functional limit theorem are extended to equivalences between the two types by applying well-known results about processes drifting to infinity and the corresponding inverse processes.     

K-flows are forward deterministic,backward completely non-deterministic stationary point processes

Stationary processes with discrete time parameter and finitely many states are forward deterministic if and only if they are backward deterministic. In contrast to this we prove in the case of continuous time parameter: Every K-flow in a Lebesgue space is isomorphic to the flow of shifts of a stationary forward deterministic, backward completely non-deterministic process (X_t, t?R¹) with two states and with right-continuous paths having only finitely many jumps in any finite time-interval. The process may be considered as a point process. The result is obtained from a representation theorem for flows, describing increasing sub-σ-algebras of a flow up to “equivalence.”     

The maximum term of uniformly mixing stationary processes

Let {X _n } be a uniformly (or strongly) mixing stationary process and let Z _n =max(X ₁, X ₂,..., X _n ). For >0, let c _n ()=inf {xR: n P(X ₁>x)}. Under a condition which holds for all -mixing processes, necessary and sufficient conditions are given for P(Z_nc_n()) to converge to each possible limit. Some conditions for convergence of P(Z_nd_n) for any sequence d _n are also obtained.Research supported in part by the National Research Council of Canada and done at the Summer Research Institute of the Canadian Mathematical Congress.We are grateful to Professor D.L. McLeish and the referee for some useful comments, particularly in connection with Lemma 2.