## The Annals of Applied Probability

### On the variance of the number of maxima in random vectors and its applications

#### Abstract

We derive a general asymptotic formula for the variance of the number of maxima in a set of independent and identically distributed random vectors in $\mathbb{R}^d$, where the components of each vector are independently and continuously distributed. Applications of the results to algorithmic analysis are also indicated.

#### Article information

Source
Ann. Appl. Probab., Volume 8, Number 3 (1998), 886-895.

Dates
First available in Project Euclid: 9 August 2002

https://projecteuclid.org/euclid.aoap/1028903455

Digital Object Identifier
doi:10.1214/aoap/1028903455

Mathematical Reviews number (MathSciNet)
MR1627803

Zentralblatt MATH identifier
0941.60021

#### Citation

Bai, Zhi-Dong; Chao, Chern-Ching; Hwang, Hsien-Kuei; Liang, Wen-Qi. On the variance of the number of maxima in random vectors and its applications. Ann. Appl. Probab. 8 (1998), no. 3, 886--895. doi:10.1214/aoap/1028903455. https://projecteuclid.org/euclid.aoap/1028903455

#### References

• Bailey, D. D., Borwein, J. M. and Girgensohn, R. (1994). Experimental evaluation of Euler sums. Experiment. Math. 3 17-30.
• Barndorff-Nielsen, O. and Sobel, M. (1966). On the distribution of the number of admissible points in a vector random sample. Theory Probab. Appl. 11 249-269.
• Becker, R. A., Denby, L., McGill, R. and Wilks, A. R. (1987). Analy sis of data from Places Rated Almanac. Amer. Statist. 41 169-186.
• Bentley, J. L., Clarkson, K. L. and Levine, D. B. (1993). Fast linear expected-time algorithms for computing maxima and convex hulls. Algorithmica 9 168-183.
• Bentley, J. L., Kung, H. T., Schkolnick, M. and Thompson, C. D. (1978). On the average number of maxima in a set of vectors and applications. J. Assoc. Comput. Mach. 25 536-543.
• Bentley, J. L. and Shamos, M. I. (1978). Divide and conquer for linear expected time. Inform. Process. Lett. 7 87-91.
• Berezovskii, B. A. and Travkin, S. I. (1975). Supervision of queues of requests in computer sy stems. Automat. Remote Control 36 1719-1725.
• Buchta, C. (1989). On the average number of maxima in a set of vectors. Inform. Process. Lett. 33 63-65.
• Devroy e, L. (1980). A note on finding convex hulls via maximal vectors. Inform. Process. Lett. 11 53-56.
• Devroy e, L. (1983). Moment inequalities for random variables in computational geometry. Computing 30 111-119.
• Devroy e, L. (1997). A note on the expected time for finding maxima by list algorithms. Algorithmica. To appear.
• Flajolet, P. and Golin, M. (1993). Exact asy mptotics of divide-and-conquer recurrences. Lecture Notes in Comput. Sci. 700 137-149. Springer, Berlin.
• Flajolet, P. and Salvy, B. (1996). Euler sums and contour integral representations. Experiment. Math. To appear.
• Golin, M. J. (1993). How many maxima can there be? Comput. Geom. 2 335-353.
• Hoffman, M. E. (1992). Multiple harmonic sums. Pacific J. Math. 152 275-290. Ivanin, V. M. (1975a). Asy mptotic estimate for the mathematical expectation of the number of elements in the Pareto set. Cy bernetics 11 108-113. Ivanin, V. M. (1975b). Estimate of the mathematical expectation of the number of elements in a Pareto set. Cy bernetics 11 506-507.
• Ivanin, V. M. (1976). Calculation of the dispersion of the number of elements of the Pareto set for the choice of independent vectors with independent components. In Theory of Optimal Decisions 90-100. Akad. Nauk. Ukrain. SSR Inst. Kibernet., Kiev. (In Russian.)
• O'Neill, B. (1980). The number of outcomes in the Pareto-optimal set of discrete bargaining games. Math. Oper. Res. 6 571-578.
• Preparata, F. P. and Shamos, M. I. (1985). Computational Geometry: An Introduction. Springer, New York.
• Sholomov, L. A. (1983). Survey of estimational results in choice problems. Engrg. Cy bernetics 21 51-75.
• Zagier, D. (1992). Values of zeta functions and their applications. In First European Congress of Mathematics 2 497-512. Birkh¨auser, Berlin.