## Electronic Journal of Probability

### Asymptotic Moments of Near Neighbor Distances for the Gaussian Distribution

Elia Liitiäinen

#### Abstract

We study the moments of the k-th nearest neighbor distance for independent identically distributed points in $\mathbb{R}^n$. In the earlier literature, the case with power higher than n has been analyzed by assuming a bounded support for the underlying density. The boundedness assumption is removed by assuming the multivariate Gaussian distribution. In this case, the nearest neighbor distances show very different behavior in comparison to earlier results.

#### Article information

Source
Electron. J. Probab., Volume 16 (2011), paper no. 92, 2545-2573.

Dates
Accepted: 3 December 2011
First available in Project Euclid: 1 June 2016

https://projecteuclid.org/euclid.ejp/1464820261

Digital Object Identifier
doi:10.1214/EJP.v16-969

Mathematical Reviews number (MathSciNet)
MR2869415

Zentralblatt MATH identifier
1245.60012

Rights

#### Citation

Liitiäinen, Elia. Asymptotic Moments of Near Neighbor Distances for the Gaussian Distribution. Electron. J. Probab. 16 (2011), paper no. 92, 2545--2573. doi:10.1214/EJP.v16-969. https://projecteuclid.org/euclid.ejp/1464820261

#### References

• D. Evans, A. J. Jones. A proof of the gamma test. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 458 (2002), no. 2027, 2759–2799.
• D. Evans, A. J. Jones, W. M. Schmidt. Asymptotic moments of near-neighbour distance distributions. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 458 (2002), no. 2028, 2839–2849.
• J. Hansen. Theory of Simple Liquids. Academic Press, 2006. ISBN: 0-12-370535-5.
• M. Kohler, A. Krzyzak, H. Walk. Rates of convergence for partitioning and nearest neighbor regression estimates with unbounded data. J. Multivariate Anal. 97 (2006), no. 2, 311–323.
• L. F. Kozachenko, N. N. Leonenko. Sample estimate of entropy of a random vector. Problems of Information Transmission 23 (1987), no. 2, 9–16.
• N. Leonenko, L. Pronzato. Correction: A class of Rényi information estimators for multidimensional densities [ ]. Ann. Statist. 38 (2010), no. 6, 3837–3838.
• N. Leonenko, L. Pronzato, V. Savani. A class of Rényi information estimators for multidimensional densities. Ann. Statist. 36 (2008), no. 5, 2153–2182.
• E. Liitiäinen, A. Lendasse, F. Corona. A boundary corrected expansion of the moments of nearest neighbor distributions. Random Structures Algorithms 37 (2010), no. 2, 223–247.
• E. Liitiäinen, A. Lendasse, F. Corona. On the statistical estimation of Rényi entropies. IEEE Conference on Machine Learning for Signal Processing (2009).
• M. Penrose. Random geometric graphs. Oxford Studies in Probability, 5. Oxford University Press, Oxford, 2003. xiv+330 pp. ISBN: 0-19-850626-0
• M. D. Penrose. Laws of large numbers in stochastic geometry with statistical applications. Bernoulli 13 (2007), no. 4, 1124–1150.
• M. D. Penrose, J. E. Yukich. Laws of large numbers and nearest neighbor distances. Advances in directional and linear statistics, 189–199, Physica-Verlag/Springer, Heidelberg, 2011.
• B. Ranneby, S. R. Jammalamadaka, A. Teterukovskiy. The maximum spacing estimation for multivariate observations. J. Statist. Plann. Inference 129 (2005), no. 1-2, 427–446.
• A. R. Wade. Explicit laws of large numbers for random nearest-neighbour-type graphs. Adv. in Appl. Probab. 39 (2007), no. 2, 326–342.
• J. E. Yukich. Probability theory of classical Euclidean optimization problems. Lecture Notes in Mathematics, 1675. Springer-Verlag, Berlin, 1998. x+152 pp. ISBN: 3-540-63666-8