×

zbMATH — the first resource for mathematics

Estimation of limiting conditional distributions for the heavy tailed long memory stochastic volatility process. (English) Zbl 1273.60028
Consider a stochastic volatility process defined as \[ Y_{i}=\sigma\left( X_{i}\right) Z_{i},\mathstrut i\in\mathbb{Z}\text{,} \] where \(\sigma\) is some (possibly unknown) positive function, \(\left\{ Z_{j},j\in\mathbb{Z}\right\} \) is an i.i.d. sequence and \(\left\{ X_{j} ,j\in\mathbb{Z}\right\} \) is a stationary Gaussian process with mean zero, unit variance, and independent from the process \(\left\{ X_{j}\right\} \). The authors study certain extremal properties of the finite dimensional joint distributions of the process \(\left\{ Y_{j}\right\} \) when \(Z_{1}\) is heavy tailed and the Gaussian process \(\left\{ X_{j}\right\} \) possibly has long memory.
For fixed positive integers \(h<m\) and \(h^{\prime}\geqq0\), Borel sets \(A\subset\mathbb{R}^{h}\) and \(B\subset\mathbb{R}^{h^{\prime}+1}\), they are interested in the limits \[ \rho\left( A,B,m\right) =\lim\limits_{t\rightarrow\infty}\mathbb{P}\left( \left( Y_{m},\dots,Y_{m+h^{\prime}}\right) \in B\mid\left( Y_{1} ,\dots,Y_{h}\right) \in tA\right) \text{.} \]
The general aim of this paper is to investigate the existence of the limiting conditional distributions \(\rho\left( A,B,m\right)\) and their statistical estimation. The asymptotic properties of estimators are studied.

MSC:
60F05 Central limit and other weak theorems
60G70 Extreme value theory; extremal stochastic processes
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Arcones, MA, Limit theorems for nonlinear functionals of a stationary Gaussian sequence of vectors, Ann. Probab., 22, 2242-2274, (1994) · Zbl 0839.60024
[2] Bickel, PJ; Wichura, MJ, Convergence criteria for multiparameter stochastic processes and some applications, Ann. Math. Stat., 42, 1656-1670, (1971) · Zbl 0265.60011
[3] Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968) · Zbl 0172.21201
[4] Bingham, NH; Goldie, CM; Teugels, JL, Regular variation, (1989), Cambridge · Zbl 0667.26003
[5] Breidt, FJ; Crato, N; Lima, P, The detection and estimation of long memory in stochastic volatility, J. Econom., 83, 325-348, (1998) · Zbl 0905.62116
[6] Breiman, L, On some limit theorems similar to the arc-sine law, Theory Probab. Appl., 10, 323-331, (1965) · Zbl 0147.37004
[7] Das, B; Resnick, SI, Conditioning on an extreme component: model consistency with regular variation on cones, Bernoulli, 17, 226-252, (2011) · Zbl 1284.60103
[8] Davis, RA; Mikosch, T, The extremogram: a correlogram for extreme events, Bernoulli, 38A, 977-1009, (2009) · Zbl 1200.62104
[9] Drees, H, Optimal rates of convergence for estimates of the extreme value index, Ann. Stat., 26, 434-448, (1998) · Zbl 0934.62047
[10] Harvey, AC; Knight, J (ed.); Satchell, S (ed.), Long memory in stochastic volatility, (1998), London
[11] Hurvich, CM; Moulines, E; Soulier, P, Estimating long memory in volatility, Econometrica, 73, 1283-1328, (2005) · Zbl 1151.91702
[12] Kulik, R; Soulier, P, The tail empirical process for long memory stochastic volatility sequences, Stoch. Process. Their Appl., 121, 109-134, (2011) · Zbl 1253.60030
[13] Mitra, A; Resnick, SI, Hidden regular variation and detection of hidden risks, Stoch. Models, 27, 591-614, (2011) · Zbl 1230.91080
[14] Omey, E; Willekens, E, Second-order behaviour of distributions subordinate to a distribution with finite Mean, Commun. Stat. Stoch. Models, 3, 311-342, (1987) · Zbl 0635.60018
[15] Orey, S, A central limit theorem for \(m\)-dependent random variables, Duke Math. J., 25, 543-546, (1958) · Zbl 0107.13403
[16] Resnick, SI, Heavy-tail phenomena, (2007), New York · Zbl 1152.62029
[17] Resnick, SI, Multivariate regular variation on cones: application to extreme values, hidden regular variation and conditioned limit laws, Stochastics, 80, 269-298, (2008) · Zbl 1142.60042
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.