## Bernoulli

• Bernoulli
• Volume 13, Number 3 (2007), 754-781.

### Asymptotic normality for the counting process of weak records and δ-records in discrete models

#### Abstract

Let {Xn, n≥1} be a sequence of independent and identically distributed random variables, taking non-negative integer values, and call Xn a δ-record if Xn>max {X1, …, Xn−1}+δ, where δ is an integer constant. We use martingale arguments to show that the counting process of δ-records among the first n observations, suitably centered and scaled, is asymptotically normally distributed for δ≠0. In particular, taking δ=−1 we obtain a central limit theorem for the number of weak records.

#### Article information

Source
Bernoulli, Volume 13, Number 3 (2007), 754-781.

Dates
First available in Project Euclid: 7 August 2007

https://projecteuclid.org/euclid.bj/1186503485

Digital Object Identifier
doi:10.3150/07-BEJ6027

Mathematical Reviews number (MathSciNet)
MR2348749

Zentralblatt MATH identifier
1138.60309

#### Citation

Gouet, Raúl; López, F. Javier; Sanz, Gerardo. Asymptotic normality for the counting process of weak records and δ -records in discrete models. Bernoulli 13 (2007), no. 3, 754--781. doi:10.3150/07-BEJ6027. https://projecteuclid.org/euclid.bj/1186503485

#### References

• Ahsanullah, M. (1995). Record Statistics. Commack: Nova Science Publishers.
• Arnold, B.C., Balakrishnan, N. and Nagaraja, H.N. (1998). Records. New York: Wiley.
• Bai, Z., Hwang, H. and Liang, W. (1998). Normal approximation of the number of records in geometrically distributed random variables. Random Structures Algorithms 13 319–334.
• Balakrishnan, N., Balasubramanian, K. and Panchapakesan, S. (1996). $\delta$-exceedance records. J. Appl. Statist. Sci. 4 123–132.
• Balakrishnan, N., Pakes, A.G. and Stepanov, A. (2005). On the number and sum of near-record observations. Adv. in Appl. Probab. 37 765–780.
• Balakrishnan, N. and Stepanov, A. (2004). A note on the paper of Khmaladze et al. Statist. Probab. Lett. 68 415–419.
• Deheuvels, P. (1974). Valeurs extrémales d'échantillons croissants d'une variable aléatoire réelle. Ann. Inst. H. Poincaré Sect. B (N.S.) 10 89–114.
• Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events. Heidelberg: Springer.
• Gouet, R., López, F.J. and San Miguel, M. (2001). A martingale approach to strong convergence of the number of records. Adv. in Appl. Probab. 33 864–873.
• Gouet, R., López, F.J. and Sanz, G. (2005). Central limit theorems for the number of records in discrete models. Adv. in Appl. Probab. 37 781–800.
• Hall, P. and Heyde, C.C. (1980). Martingale Limit Theory and Its Application. New York: Academic Press.
• Hashorva, E. (2003). On the number of near-maximum insurance claim under dependence. Insurance Math. Econom. 32 37–49.
• Hashorva, E. and Hüsler, J. (2005). Estimation of tails and related quantities using the number of near-extremes. Comm. Statist. Theory Methods 34 337–349.
• Key, E.S. (2005). On the number of records in an i.i.d. discrete sequence. J. Theoret. Probab. 18 99–107.
• Khmaladze, E., Nadareishvili, M. and Nikabadze, A. (1997). Asymptotic behaviour of a number of repeated records. Statist. Probab. Lett. 35 49–58.
• Li, Y. (1999). A note on the number of records near the maximum. Statist. Probab. Lett. 43 153–158.
• Neveu, J. (1975). Discrete-Parameter Martingales. Amsterdam: North-Holland.
• Nevzorov, V.B. (2001). Records: Mathematical Theory. Translations of Mathematical Monographs 194. Providence, RI: American Mathematical Society.
• Pakes, A.G. (2000). The number and sum of near-maxima for thin-tailed populations. Adv. in Appl. Probab. 32 1100–1116.
• Pakes, A.G. and Steutel, F.W. (1997). On the number of records near the maximum. Austral. J. Statist. 39 179–192.
• Renyi, A. (1962). Théorie des éléments saillants d'une suite d'observations. Ann. Fac. Sci. Univ. Clermont-Ferrand 8 7–13.
• Vervaat, W. (1973). Limit theorems for records from discrete distributions. Stochastic Processes Appl. 1 317–334.