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

Abstract

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.