Limit distributions of extremal distances to the nearest neighbor

A. M. Zubkov ; O. P. Orlov

Diskretnaya Matematika, Tome 29 (2017) no. 2, pp. 3-17.

Résumé

Theorems on the limit distributions of the minimal and maximal distances to the nearest neighbor in a sample of random independent points having a uniform distribution on a metric space are proved. As examples of such spaces a multidimensional torus and a binary cube are considered.

Détail

Keywords: random points in a metric space, nearest neighbors, distributions of extremal values, binary cube.

Bibliographie

[1] Sevast'yanov B. A., “Poisson limit law for a scheme of sums of dependent random variables”, Theory Probab. Appl., 17:4 (1973), 695–699 | DOI | MR | Zbl

[2] Silverman B., Brown T., “Short distances, flat triangles and Poisson limits”, J. Appl. Probab., 15 (1978), 815–825 | DOI | MR | Zbl

[3] Henze N., “The limit distribution for maxima of «weighted» $r$-th nearest-neighbor distances”, J. Appl. Prob., 19 (1982), 344–354 | MR | Zbl

[4] Penrose M. D., Yukich J. E., “Laws of large numbers and nearest neighbor distances”, Advances in Directional and Linear Statistics, Physica-Verlag HD, Berlin, 2011, 189–199 | DOI | MR

[5] Baryshnikov Yu., Penrose M. D., Yukich J. E., “Gaussian limits for generalized spacings”, Ann. Appl. Probab., 19:1 (2009), 158–185 | DOI | MR | Zbl

[6] Bickel P. J., Breiman L., “Sums of functions of nearest neighbor distances, moment bounds, limit theorems and a goodness of fit test”, Ann. Probab., 11:1 (1983), 185–214 | DOI | MR | Zbl

[7] Schilling M. F., “Goodness of fit testing in $R^m$ based on the weighted empirical distribution of certain nearest neighbor statistics”, Ann. Statist., 11:1 (1983), 1–12 | DOI | MR | Zbl

[8] Schilling M. F., “An infinite-dimensional approximation for nearest neighbor goodness of fit tests”, Ann. Statist., 11:1 (1983), 13–24 | DOI | MR | Zbl

[9] L'écuyer P., Cordeau J.-F., Simard R., “Close-point spatial tests and their application to random number generators”, Oper. Res., 48:2 (2000), 308–317 | DOI | MR

[10] Mikhailov V. G., “A Central limit theorem for the number of partial long repetitions”, Theory Probab. Appl., 20:4 (1976), 862–866 | DOI | MR

[11] Mikhailov V. G., “Poisson-type limit theorems for the number of incomplete matches of S-patterns”, Theory Probab. Appl., 47:2 (2003), 343–351 | DOI | DOI | MR

[12] Burden C. J., Kantorovitz M. R., Wilson S. R., “Approximate word matches between two random sequences”, Ann. Appl. Probab., 18:1 (2008), 1–21 | DOI | MR | Zbl

[13] Zubkov A. M., Mikhailov V. G., “Limit distributions of random variables associated with long duplications in a sequence of independent trials”, Theory Probab. Appl., 19:1 (1974), 172–179 | DOI | MR | Zbl

[14] Karlin S., Ost F., “Counts of long aligned word matches among random letter sequences”, Adv. Appl. Prob., 19:2 (1987), 293–351 | DOI | MR | Zbl

[15] Zubkov A. M., Kruglov V. I., “On coincidences of tuples in a binary tree with random labels of vertices”, Discrete Math. Appl., 26:3 (2016), 145–153 | DOI | DOI | MR | Zbl

[16] Mikhaylov V. G., “Estimates of accuracy of the Poisson approximation for the distribution of number of runs of long string repetitions in a Markov chain”, Discrete Math. Appl., 26:2 (2016), 105–113 | DOI | DOI | MR | Zbl

[17] Mikhailov V. G., “O veroyatnosti nalichiya v sluchainoi posledovatelnosti tsepochek s odinakovoi strukturoi”, Diskretnaya matematika, 28:3 (2016), 97–110 | DOI

[18] Burago Yu. D., Zalgallier V. A., Geometric inequalities, Springer, Berlin – N. Y., 1988 | MR | MR | Zbl

[19] Riordan J., An Introduction to Combinatorial Analysis, John Wiley Sons, 1958, 256 pp. | MR | Zbl