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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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.

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

First available in Project Euclid: 31 March 2015

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Zentralblatt MATH identifier

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

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


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.

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  • Anantharam, V. and Baccelli, F. (2008). A Palm theory approach to error exponents. In Proc. IEEE Internat. Symp. Inf. Theory, IEEE, pp. 1768–1772.
  • Anantharam, V. and Baccelli, F. (2010). Information-theoretic capacity and error exponents of stationary point processes under random additive displacements. Preprint. Available at
  • Ashikhmin, A. E., Barg, A. and Litsyn, S. N. (2000). A new upper bound on the reliability function of the Gaussian channel. IEEE Trans. Inf. Theory 46, 1945–1961.
  • Barron, A. (1985). The strong Ergodic theorem for densities: generalized Shannon–McMillan–Breiman theorem, Ann. Prob., 13, 1292–1303.
  • Cover, T. M. and Thomas, J. A. (1991). Elements of Information Theory. John Wiley, New York.
  • Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.
  • Dembo, A. and Zeitouni, O. (1993). Large Deviations Techniques and Applications. Jones and Bartlett, Boston, MA.
  • El Gamal, A. and Kim, Y.-H. (2011). Network Information Theory. Cambridge University Press.
  • Gallager, R. G. (1968). Information Theory and Reliable Communication. John Wiley, New York.
  • Gray, R. M. (2006). Toeplitz and Circulant Matrices: A Review. NOW Publishers, Delft.
  • Han, T. S. (2003). Information-Spectrum Methods in Information Theory. Springer, Berlin.
  • Kallenberg, O. (1983). Random Measures, 3rd edn. Akademie-Verlag, Berlin.
  • Kieffer, J. C. (1974). A simple proof of the Moy–Perez generalization of the Shannon–McMillan theorem. Pacific J. Math. 51, 203–206.
  • Matérn, B. (1960). Spatial Variation, 2nd edn. Springer, Berlin.
  • Møller, J. (1994). Lectures on Random Voronoĭ Tessellations (Lecture Notes Statist. 87). Springer, New York.
  • Poltyrev, G. (1994). On coding without restrictions for the AWGN channel. IEEE Trans. Inf. Theory 40, 409–417.
  • Shannon, C. E. (1948). A mathematical theory of communication. Bell Sys. Tech. J. 27, 379–423, 623–656.
  • Shannon, C. E. (1959). Probability of error for optimal codes in a Gaussian channel. Bell Sys. Tech. J. 38, 611–656.
  • Varadhan, S. R. S. (1984). Large Deviations and Applications. SIAM, Philadelphia, PA.