Advances in Applied Probability

Capacity and error exponents of stationary point processes under random additive displacements

Venkat Anantharam and François Baccelli

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Abstract

Consider a real-valued discrete-time stationary and ergodic stochastic process, called the noise process. For each dimension n, we can choose a stationary point process in Rn and a translation invariant tessellation of Rn. Each point is randomly displaced, with a displacement vector being a section of length n of the noise process, independent from point to point. The aim is to find a point process and a tessellation that minimizes the probability of decoding error, defined as the probability that the displaced version of the typical point does not belong to the cell of this point. We consider the Shannon regime, in which the dimension n tends to ∞, while the logarithm of the intensity of the point processes, normalized by dimension, tends to a constant. We first show that this problem exhibits a sharp threshold: if the sum of the asymptotic normalized logarithmic intensity and of the differential entropy rate of the noise process is positive, then the probability of error tends to 1 with n for all point processes and all tessellations. If it is negative then there exist point processes and tessellations for which this probability tends to 0. The error exponent function, which denotes how quickly the probability of error goes to 0 in n, is then derived using large deviations theory. If the entropy spectrum of the noise satisfies a large deviations principle, then, below the threshold, 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 obtained for two classes of point processes: the Poisson process and a Matérn hard-core point process. New lower bounds on error exponents are derived from this for Shannon's additive noise channel in the high signal-to-noise ratio limit that hold for all stationary and ergodic noises with the above properties and that match the best known bounds in the white Gaussian noise case.

Article information

Source
Adv. in Appl. Probab., Volume 47, Number 1 (2015), 1-26.

Dates
First available in Project Euclid: 31 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.aap/1427814578

Digital Object Identifier
doi:10.1239/aap/1427814578

Mathematical Reviews number (MathSciNet)
MR3327312

Zentralblatt MATH identifier
1318.60055

Subjects
Primary: 60G55: Point processes 94A15: Information theory, general [See also 62B10, 81P94]
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60F10: Large deviations

Keywords
Point process random tessellation high-dimensional stochastic geometry information theory entropy spectrum large deviations theory

Citation

Anantharam, Venkat; Baccelli, François. Capacity and error exponents of stationary point processes under random additive displacements. Adv. in Appl. Probab. 47 (2015), no. 1, 1--26. doi:10.1239/aap/1427814578. https://projecteuclid.org/euclid.aap/1427814578


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