Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-24T20:06:49.957Z Has data issue: false hasContentIssue false
  • English
  • Français

The asymptotic joint normality of the numbers of upper records, lower records and inversions in a random sequence

Part of: Limit theorems Combinatorics

Published online by Cambridge University Press:  14 July 2016

Chern-Ching Chao ,
Yi-Liang Chen  and
Wei-Hou Cheng
Chern-Ching Chao*
Affiliation:
Academia Sinica, Taipei
Yi-Liang Chen*
Affiliation:
Aletheia University
Wei-Hou Cheng*
Affiliation:
Tamkang University
*
∗ Postal address: Institute of Statistical Science, Academia Sinica, Taipei, Taiwan 115, Republic of China.
∗∗ Postal address: Department of Mathematical Statistics and Actuarial Science, Aletheia University, Tamsui, Taiwan 251, Republic of China.
∗∗∗ Postal address: Department of Mathematics, Tamkang University, Tamsui, Taiwan 251, Republic of China. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We derive the asymptotic joint normality, by a martingale approach, for the numbers of upper records, lower records and inversions in a random sequence.

Keywords

Asymptotic joint normalitymartingale central limit theorem

MSC classification

Primary: 05A16: Asymptotic enumeration 60F05: Central limit and other weak theorems
Type
Short Communications
Information
Journal of Applied Probability , Volume 40 , Issue 1 , March 2003 , pp. 257 - 263
Copyright
Copyright © Applied Probability Trust 2003 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.Google Scholar
Brunk, H. D. (1960). On a theorem of E. Sparre Anderson and its application to tests against trend. Math. Scand. 4, 305–326.Google Scholar
David, F. N., and Barton, D. E. (1962). Combinatorial Chance. Griffin, London.Google Scholar
Foster, F. G., and Stuart, A. (1954). Distribution-free tests in time-series based on the breaking of records. J. R. Statist. Soc. B 16, 1–22.Google Scholar
Haghighi-Talab, D., and Wright, C. (1973). On the distribution of records in a finite sequence of observations, with an application to road traffic problems. J. Appl. Prob. 10, 556–571.Google Scholar
Helland, I. S. (1982) Central limit theorems for martingales with discrete or continuous time. Scand. J. Statist. 9, 79–94.Google Scholar
Karlin, S., and Taylor, H. M. (1981). A Second Course in Stochastic Processes. Academic Press, New York.Google Scholar
Katzenbeisser, W. (1988). On the joint distribution of the random variables ‘number of inversions’ and ‘number of outstanding variables’ in a randomly arranged sequence. Statist. Hefte 29, 133–141.Google Scholar
Katzenbeisser, W. (1990). On the joint distribution of the number of upper and lower records and the number of inversions in a random sequence. Adv. Appl. Prob. 22, 957–960.Google Scholar
McLeish, D. L. (1974). Dependent central limit theorems and invariance principles. Ann. Prob. 2, 620–628.Google Scholar
Pfeifer, D. (1989). Extremal processes, secretary problems and the 1/e law. J. Appl. Prob. 26, 722–733.CrossRefGoogle Scholar
Rényi, A. (1962). Théorie des éléments saillants d'une suite d'observations. In Proc. Colloq. Combinatorial Methods in Probability Theory, Mathematics Institute, Aarhus University, pp. 104–115.Google Scholar
Resnick, S. I. (1973). Record values and maxima. Ann. Prob. 1, 650–662.Google Scholar
Shorrock, R. (1972). On record values and record times. J. Appl. Prob. 9, 316–326.CrossRefGoogle Scholar
Shorrock, R. (1973). Record values and inter-record times. J. Appl. Prob. 10, 543–555.Google Scholar
Shorrock, R. (1974). On discrete time extremal processes. Adv. Appl. Prob. 6, 580–592.Google Scholar
Sparre Anderson, E. (1954). On the fluctuation of sums of random variables. II. Math. Scand. 2, 195–223.Google Scholar
Takács, L. (1984). Combinatorics. In Handbook of Statistics, Vol. 4, Nonparametric Methods, eds Krishnaiah, P. R. and Sen, P. K., North Holland, Amsterdam, pp. 123–143.Google Scholar