Capacity and Error Exponents of Stationary Point Processes under Random Additive Displacements

Abstract

This paper studies the Shannon regime for the random displacement of stationary point processes. Let each point of some initial stationary point process in R n give rise to one daughter point, the location of which is obtained by adding a random vector to the coordinates of the mother point, with all displacement vectors independently and identically distributed for all points. The decoding problem is then the following one: the whole mother point process is known as well as the coordinates of some daughter point; the displacements are only known through their law; can one find the mother of this daughter point? The Shannon regime is that where the dimension n tends to infinity and where the logarithm of the intensity of the point process is proportional to n. We show that this problem exhibits a sharp threshold: if the sum of the proportionality factor and of the differential entropy rate of the noise is positive, then the probability of finding the right mother point tends to 0 with n for all point processes and decoding strategies. If this sum is negative, there exist mother point processes, for instance Poisson, and decoding strategies, for instance maximum likelihood, for which the probability of finding the right mother tends to 1 with n. We then use large deviations theory to show that in the latter case, if the entropy spectrum of the noise satisfies a large deviation principle, then the error probability goes exponentially fast to 0 with an exponent that is given in closed form in terms of the rate function of the noise entropy spectrum. This is done for two classes of mother point processes: Poisson and Matérn.This class of problems is of independent relevance in point process theory, where this Shannon regime has hardly been considered. Its practical interest to information theory comes from the explicit connection that we also establish between this problem and the estimation of error exponents in Shannon's additive noise channel with power constraints on the codewords. Each error exponent for the point process problem leads to an

The Boolean model in the Shannon regime: three thresholds and related asymptotics

“…its law under P 0 n ) is equivalent in law to the superposition of a stationary version of the process and a process with a single point at the origin carrying a ball with radius having lawX n √ n, and independent of the stationary version (which is called the reduced process of the Palm version). Our motivations for the analysis of this setting came from related problems in information theory that we studied recently [1]. More specifically, in the study of error probabilities for coding over an additive white Gaussian noise channel [4,Section 7.4], it is natural to consider a sequence of Poisson processes, one in each dimension n ≥ 1, with well defined asymptotic logarithmic intensity, as was done in [1], motivated by the ideas in [9].…”

“…Our motivations for the analysis of this setting came from related problems in information theory that we studied recently [1]. More specifically, in the study of error probabilities for coding over an additive white Gaussian noise channel [4,Section 7.4], it is natural to consider a sequence of Poisson processes, one in each dimension n ≥ 1, with well defined asymptotic logarithmic intensity, as was done in [1], motivated by the ideas in [9]. The error exponent questions studied in [1] are related to 1 the consideration of a Boolean model where the grains associated with the individual points are defined in terms of additive white Gaussian noise: for all n ≥ 1 and k ≥ 1, let W (i,k) n , n ≥ i ≥ 1, denote an i.i.d.…”

“…This fits within the class of deterministic Boolean models considered in this paper. Thus, having carried out the analysis in [1], it was natural for us to become curious about the asymptotic properties in dimension of the sequence of Boolean 1 The scenario considered in [1] goes beyond additive white Gaussian noise to consider a setting where the additive noise comes from sections of a stationary and ergodic process. The Boolean models that arise in the more general case involve grains, defined by the typicality sets of the noise process, that are not necessarily spherically symmetric.…”

“…The Boolean model was considered in high dimensions in a few papers, both within the framework of stochastic geometry [5,8] and within the framework of information theory [1]. The present paper discusses three thresholds and some asymptotics related to these thresholds in a setting analogous to that in [1], which is that where the radii of the balls in the Boolean model scale with the dimension n of the ambient space likeX n √ n, where (X n , n ≥ 1) is a sequence of random variables. In this paper, we assume that this sequence of random variables satisfies a large deviations principle (LDP).…”

Thin-shell concentration for zero cells of stationary Poisson mosaics

“…Recently, there has been more interest in high dimensional random tessellations, partially due to applications in signal processing [13] and information theory [2]. For these applications, it is important to understand the asymptotic geometric properties of the convex polytopes induced by random tessellations, in order to decode and reconstruct high dimensional signals with small error.…”