Limit distributions of the maximal distance to the nearest neighbour
-
Oleg P. Orlov
Abstract
For sets of iid random points having a uniform (in a definite sense) distribution on the arbitrary metric space a maximal distance to the nearest neighbour is considered. By means of the Chen–Stein method new limit theorems for this random variable is proved. For random uniform samples from the set of binary cube vertices analogous results are obtained by the methods of moments.
Originally published in Diskretnaya Matematika (2018) 30, №3, 88–98 (in Russian).
Acknowledgment
The author thanks his scientific supervisor A. M. Zubkov for the problem statement, constant attention and useful comments.
References
[1] Barbour A. D., Chen Louis H. Y., An introduction to Stein’s method, Singapore Univ. Press and World Scientific Publ. Co. Pte. Ltd, 2005.10.1142/5792Suche in Google Scholar
[2] Barbour A. D., Holst L., Janson S., Poisson approximation, Clarendon press, Oxford, 1992.Suche in Google Scholar
[3] Zubkov A. M., Orlov O. P., “Limit distributions of extremal distances to the nearest neighbor”, Discrete Math. Appl., 28:3 (2018), 189–199.10.1515/dma-2018-0018Suche in Google Scholar
[4] Hoeffding W., “Probability inequalities for sums of bounded random variables”, J. Amer. Stat. Assoc., 58:301 (1963), 13–30.10.1080/01621459.1963.10500830Suche in Google Scholar
[5] Henze N., “The limit distribution for maxima of "weighted" rth-nearest-neighbour distances”, J. Appl. Probab., 19:2 (1982), 344–354.10.2307/3213486Suche in Google Scholar
© 2019 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Periodic properties of pushdown automata
- Burnside-type problems in discrete geometry
- On affine classification of permutations on the space GF(2)3
- Limit distributions of the maximal distance to the nearest neighbour
- Elementary transformations of systems of equations over quasigroups and generalized identities
- Asymptotics for the logarithm of the number of k-solution-free sets in Abelian groups
- On the distribution of multiple power series regularly varying at the boundary point
- A letter to the Editor
Artikel in diesem Heft
- Frontmatter
- Periodic properties of pushdown automata
- Burnside-type problems in discrete geometry
- On affine classification of permutations on the space GF(2)3
- Limit distributions of the maximal distance to the nearest neighbour
- Elementary transformations of systems of equations over quasigroups and generalized identities
- Asymptotics for the logarithm of the number of k-solution-free sets in Abelian groups
- On the distribution of multiple power series regularly varying at the boundary point
- A letter to the Editor
