Skip to main content

A limiting distribution for maxima of discrete stationary triangular arrays with an application to risk due to avalanches


In this paper, we generalize earlier work dealing with maxima of discrete random variables. We show that row-wise stationary block maxima of a triangular array of integer valued random variables converge to a Gumbel extreme value distribution if row-wise variances grow sufficiently fast as the row-size increases. As a by-product, we derive analytical expressions of normalising constants for most classical unbounded discrete distributions. A brief simulation illustrates our theoretical result. Also, we highlight its usefulness in practice with a real risk assessment problem, namely the evaluation of extreme avalanche occurrence numbers in the French Alps.

This is a preview of subscription content, access via your institution.


  • Anderson, C.W.: Local limit theorems for the maxima of discrete random variables. Math. Proc. Camb. Philos. Soc. 88, 161–165 (1980)

    Article  MATH  Google Scholar 

  • Anderson, C.W., Coles, S.G., Hüsler, J.: Maxima of poisson-like variables and related triangular arrays. Ann. Appl. Probab. 7(4), 953–971 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  • Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J.: Statistics of extremes: Theory and applications. Wiley series in probability and statistics (2004)

  • Castebrunet, H., Eckert, N., Giraud, G.: Snow and weather climatic control on snow avalanche occurrence fluctuations over 50 yr in the french alps. Clim. Past 8 (2012)

  • Coles, S.G.: An introduction to statistical modeling of extreme values. Springer series in statistics (2001)

  • Consul, P.C., Famoye, F.: Lagrangian probability distributions. Birkhäuser, Boston (2006)

    MATH  Google Scholar 

  • De Haan, L., Ferreira, A.: Extreme value theory: An introduction. Springer series in operations research and financial engineering (2006)

  • Eckert, N., Coleou, C., Castebrunet, H., Deschatres, M., Giraud, G., Gaume, J.: Cross-comparison of meteorological and avalanche data for characterising avalanche cycles: The example of december 2008 in the eastern part of the french alps. Cold Reg. Sci. Technol. 64, 119–136 (2010b)

    Article  Google Scholar 

  • Eckert, N., Parent, E., Kies, R., Baya, H.: A spatio-temporal modelling framework for assessing the fluctuations of avalanche occurrence resulting from climate change: application to 60 years of data in the northern french alps. Clim. Chang. 101, 515–553 (2010a)

    Article  Google Scholar 

  • Eckert, N., Parent, E., Richard, D.: Revisiting statistical topographical methods for avalanche predetermination: Bayesian modelling for runout distance predictive distribution. Cold Region Sci. Technol. 49, 88–107 (2007)

    Article  Google Scholar 

  • Embrechts, P., Klüppelberg, C., Mikosch, T.: Modelling extremal events for insurance and finance. Springer-Verlag, Berlin (1997)

    Book  MATH  Google Scholar 

  • Falk, M., Hüsler, J., Reiss, R.D.: Laws of small numbers: Extremes and rare events, 3 edn. Basel, Birkhäuser Verlag (2010)

    Google Scholar 

  • Gaume, J., Eckert, N., Chambon, G., Naaim, M., Bel, L.: Mapping extreme snowfalls in the french alps using max-stable processes. Water Resour. Res. 49(2), 1079–1098 (2013)

    Article  Google Scholar 

  • Katz, R.W., Parlange, M., Naveau, P.: Statistics of extremes in hydrology. Adv. Water Resour. 25(8–12), 1287–1304 (2002)

    Article  Google Scholar 

  • Kim, H., Park, Y.: A non-stationary integer-valued autoregressive model. Stat. Pap. 49, 485–502 (2008)

    Article  MATH  Google Scholar 

  • Leadbetter, M.R., Lindgren, G., Rootzén, H.: Extremes and related properties of random sequences and processes. Springer-Verlag, New York (1983)

    Book  MATH  Google Scholar 

  • Rousselot, M., Durand, Y., Giraud, G., Merindol, L., Daniel, L.: Analysis and forecast of extreme new-snow avalanches: a numerical study of the avalanche cycles of february 1999 in france. J. Glaciol. 56(199), 758–770 (2010)

    Article  Google Scholar 

  • Saulis, L., Statulevicius, V.: Limit Theorems for Large Deviations. Kluwer Academic Dordrecht, Boston (1991)

    Book  MATH  Google Scholar 

  • Schweizer, J., Mitterer, C., Stoffel, L.: On forecasting large and infrequent snow avalanches. Cold Regions Scie. Technol. 59(2–3), 234–241 (2009)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Pascal Sielenou Dkengne.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Dkengne, P.S., Eckert, N. & Naveau, P. A limiting distribution for maxima of discrete stationary triangular arrays with an application to risk due to avalanches. Extremes 19, 25–40 (2016).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Extreme values
  • Triangular arrays
  • Stationary processes
  • Discrete random variables
  • Snow avalanche occurrence numbers

AMS 2000 Subject Classifications

  • 60G70
  • 60G10
  • 60F10