## Advances in Applied Probability

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

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

Venkat Anantharam and François Baccelli

#### 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 **R**^{n} and a translation
invariant tessellation of **R**^{n}. 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