Bernoulli

  • Bernoulli
  • Volume 25, Number 1 (2019), 375-394.

On the longest gap between power-rate arrivals

Søren Asmussen, Jevgenijs Ivanovs, and Johan Segers

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Let $L_{t}$ be the longest gap before time $t$ in an inhomogeneous Poisson process with rate function $\lambda_{t}$ proportional to $t^{\alpha-1}$ for some $\alpha\in(0,1)$. It is shown that $\lambda_{t}L_{t}-b_{t}$ has a limiting Gumbel distribution for suitable constants $b_{t}$ and that the distance of this longest gap from $t$ is asymptotically of the form $(t/\log t)E$ for an exponential random variable $E$. The analysis is performed via weak convergence of related point processes. Subject to a weak technical condition, the results are extended to include a slowly varying term in $\lambda_{t}$.

Article information

Source
Bernoulli, Volume 25, Number 1 (2019), 375-394.

Dates
Received: March 2017
Revised: August 2017
First available in Project Euclid: 12 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.bj/1544605250

Digital Object Identifier
doi:10.3150/17-BEJ990

Mathematical Reviews number (MathSciNet)
MR3892323

Zentralblatt MATH identifier
07007211

Keywords
Gumbel distribution inhomogeneous Poisson process point processes records regular variation weak convergence

Citation

Asmussen, Søren; Ivanovs, Jevgenijs; Segers, Johan. On the longest gap between power-rate arrivals. Bernoulli 25 (2019), no. 1, 375--394. doi:10.3150/17-BEJ990. https://projecteuclid.org/euclid.bj/1544605250


Export citation

References

  • [1] Antzoulakos, D.L. (1999). On waiting time problems associated with runs in Markov dependent trials. Ann. Inst. Statist. Math. 51 323–330.
  • [2] Asmussen, S., Fiorini, P., Lipsky, L., Rolski, T. and Sheahan, R. (2008). On the distribution of total task times for tasks that must restart from the beginning if failure occurs. Math. Oper. Res. 33 932–944.
  • [3] Asmussen, S., Ivanovs, J. and Rønn-Nielsen, A. (2016). Time inhomogeneity in longest run and longest gap problems. Stochastic Process. Appl. 127 574–589.
  • [4] Asmussen, S., Lipsky, L. and Thompson, S. (2016). Markov renewal methods in restart problem in complex systems. In The Fascination of Probability, Statistics and Their Applications 501–527. Springer, Cham.
  • [5] Balakrishnan, N. and Koutras, M.V. (2002). Runs and Scans with Applications. Wiley Series in Probability and Statistics. New York: Wiley-Interscience [John Wiley & Sons].
  • [6] Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1989). Regular Variation. Cambridge Univ. Press.
  • [7] Borovkov, K. (1999). On records and related processes for sequences with trends. J. Appl. Probab. 36 668–681.
  • [8] Calka, P. and Chenavier, N. (2014). Extreme values for characteristic radii of a Poisson–Voronoi tessellation. Extremes 17 359–385.
  • [9] de Haan, L. and Verkade, E. (1987). On extreme-value theory in the presence of a trend. J. Appl. Probab. 24 62–76.
  • [10] Erdös, P. and Revesz, P. (1975). On the length of the longest head-run. In Topics in Information Theory 219–228.
  • [11] Fu, J.C. and Koutras, M.V. (1994). Distribution theory of runs: A Markov chain approach. J. Amer. Statist. Assoc. 89 1050–1058.
  • [12] Gouet, R., Lopez, J. and Sanz, G. (2015). Records from stationary observations subject to a random trend. Adv. in Appl. Probab. 47 1175–1189.
  • [13] Hall, P. (1984). Random, nonuniform distribution of line segments on a circle. Stochastic Process. Appl. 18 239–261.
  • [14] Hall, P. (1988). Introduction to the Theory of Coverage Processes. John Wiley & Sons Incorporated.
  • [15] Hüsler, J. (1987). Minimal spacings of non-uniform densities. Stochastic Process. Appl. 25 73–81.
  • [16] Kallenberg, O. (1996). Improved criteria for distributional convergence of point processes. Stochastic Process. Appl. 64 93–102.
  • [17] Kallenberg, O. (2002). Foundations of Modern Probability. Media: Springer Science & Business.
  • [18] Karlin, S. and Chen, C. (2004). $r$-scan extremal statistics of inhomogeneous Poisson processes. In IMS Lecture Notes Monogr. Ser. 45 287–290.
  • [19] Leadbetter, M.R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer.
  • [20] Molchanov, I. and Scherbakov, V. (2003). Coverage of the whole space. Adv. in Appl. Probab. 35 898–912.
  • [21] Resnick, S.I. (2008). Extreme Values, Regular Variation and Point Processes. New York: Springer.
  • [22] Smith, R.L. (1989). Extreme value analysis of environmental time series: An application to trend detection in ground-level ozone. Statist. Sci. 4 367–393. With comments and a rejoinder by the author.
  • [23] von Mises, R. (1981). Probability, Statistics and Truth. New York: Dover Publications, Inc.
  • [24] Weissman, I. (1975). Multivariate extremal processes generated by independent non-identically distributed random variables. J. Appl. Probab. 12 477–487.
  • [25] Withers, C.S. and Nadarajah, S. (2013). Asymptotic behavior of the maximum from distributions subject to trends in location and scale. Statist. Probab. Lett. 83 2143–2151.