, Volume 16, Issue 4, pp 487–505 | Cite as

The convex hull of consecutive pairs of observations from some time series models



We examine the asymptotic behavior of the number of vertices of the convex hull spanned by n consecutive pairs from a time series model. We consider data from three models, the moving average (MA) process with regularly varying noise, the stochastic volatility (SV) process with regularly varying noise and the GARCH process. The latter two processes are commonly used for modeling returns of financial assets. If \(N_n\) denotes the number of vertices of the convex hull of n consecutive pairs of observations, we show that for a SV model, \(N_n \stackrel {P}{\rightarrow } 4 \) as \(n \rightarrow \infty\), whereas for a GARCH model, \(N_n \geq 5\) with positive probability. This provides another measure that distinguishes the behavior of the extremes for SV and GARCH models. Geometrically the extreme GARCH pairs fall in butterfly-like shapes away from the axes, while the SV pairs suitably scaled drift towards the coordinate axes with increasing n. MA pairs show a similar flavor as the SV pairs except that their convex hull vertices produce segments of extreme pairs that no longer align themselves exclusively along the axes, but are also distributed along other directions, determined solely by the MA coefficients. We show that the non-degenerate limiting distribution of \( N_n \) as \(n \rightarrow \infty \) depends on the model parameters and limiting law of the ratio of the maximal and minimal observations.


Time series Extremes Convex hull Regular variation Stochastic volatility process GARCH Moving average 

AMS 2000 Subject Classifications

Primary–60G70; Secondary–62M10 62G32 52A22 60D05 


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  1. Aldous, D.J., Fristedt, J., Griffin, P.S., Pruitt, W.E.: The number of extreme points in the convex hull of a random sample. J. Appl. Prob. 28, 287–304 (1991)MathSciNetCrossRefMATHGoogle Scholar
  2. Basrak, B., Davis, R.A., Mikosch, T.: A characterization of multivariate regular variation. Ann. Appl. Probab. 3, 908–920 (2002a)MathSciNetGoogle Scholar
  3. Basrak, B., Davis, R.A., Mikosch, T.: Regular variation of GARCH processes. Stoch. Proc. Appl. 99, 95–115 (2002b)MathSciNetCrossRefMATHGoogle Scholar
  4. Bollerslev, T.: Generalized autoregressive conditional heteroskedasticity. J. Econom. 31, 307–327 (1986)MathSciNetCrossRefMATHGoogle Scholar
  5. Bougerol, P., Picard, N.: Stationarity of GARCH processes and of some nonnegative time series. J. Econom. 52, 115–127 (1992)MathSciNetCrossRefMATHGoogle Scholar
  6. Breiman, L.: On some limit theorems similar to the arc-sin law. Theory Probab. Appl. 10, 323–331 (1965)MathSciNetCrossRefGoogle Scholar
  7. Carnal, H.: Die konvexe Hülle von n rotationssymmetrisch verteilten Punkten. Z. Wahrscheinlichkeitsth. 15, 168–176 (1970)MathSciNetCrossRefMATHGoogle Scholar
  8. Davis, R.A., Mikosch, T.: The sample autocorrelations of heavy-tailed processes with applications to ARCH. Ann. Statist. 26, 2049–2080 (1998)MathSciNetCrossRefMATHGoogle Scholar
  9. Davis, R.A., Mikosch, T.: Point process convergence of stochastic volatility processes with application to sample autocorrelation. Probability, statistics and seismology. J. Appl. Probab. 38, 93–104 (2001)MathSciNetCrossRefGoogle Scholar
  10. Davis, R.A., Mikosch, T.: Extreme value theory for GARCH processes. In: Andersen, T.G., Davis, R.A., Kreiss, J.-P., Mikosch, T. (eds.) Handbook of Financial Time Series, pp. 187–200. Springer, New York (2009)CrossRefGoogle Scholar
  11. Davis, R.A., Mulrow, E., Resnick, S.I.: Almost sure limit sets of random samples in \( \textbf{R}^d\). Adv. Appl. Probab. 20, 573–599 (1988)MathSciNetCrossRefMATHGoogle Scholar
  12. Engle, R.F.: Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50, 987–1007 (1982)MathSciNetCrossRefMATHGoogle Scholar
  13. Resnick, S.I.: Heavy-Tail Phenomena Probabilistic and Statistical Modeling. Springer Verlag, New York (2007)MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of StatisticsColumbia UniversityNew YorkUSA

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