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Extremes of a class of deterministic sub-sampled processes with applications to stochastic difference equations

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  • Scotto, M.

Abstract

Let Xk k[greater-or-equal, slanted]1 be a stationary sequence of the formwhere Ak,Bk are i.i.d. -valued random pairs with some given joint distribution. For a strictly increasing subsequence g(k) , let Yk=Xg(k) be the deterministic sub-sampled sequence. The aim of this paper is to look at the limiting form of certain empirical point processes induced by Yk for a specific class of deterministic sampling functions g(·). Such asymptotic results will be useful in obtaining the weak limiting behavior of various functionals of the underlying process including the asymptotic distribution of upper and lower order statistics. In particular, we investigate the limiting distribution of the maximum and its corresponding extremal index.

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  • Scotto, M., 2005. "Extremes of a class of deterministic sub-sampled processes with applications to stochastic difference equations," Stochastic Processes and their Applications, Elsevier, vol. 115(3), pages 417-434, March.
    • Handle: RePEc:eee:spapps:v:115:y:2005:i:3:p:417-434
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    1. Dacorogna, Michael M. & Muller, Ulrich A. & Nagler, Robert J. & Olsen, Richard B. & Pictet, Olivier V., 1993. "A geographical model for the daily and weekly seasonal volatility in the foreign exchange market," Journal of International Money and Finance, Elsevier, vol. 12(4), pages 413-438, August.
    2. M. G. Scotto & K. F. Turkman & C. W. Anderson, 2003. "Extremes of Some Sub‐Sampled Time Series," Journal of Time Series Analysis, Wiley Blackwell, vol. 24(5), pages 579-590, September.
    3. Muller, Ulrich A. & Dacorogna, Michel M. & Olsen, Richard B. & Pictet, Olivier V. & Schwarz, Matthias & Morgenegg, Claude, 1990. "Statistical study of foreign exchange rates, empirical evidence of a price change scaling law, and intraday analysis," Journal of Banking & Finance, Elsevier, vol. 14(6), pages 1189-1208, December.
    4. M. E. Robinson & J. A. Tawn, 2000. "Extremal analysis of processes sampled at different frequencies," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 62(1), pages 117-135.
    5. Hall, A. & Scotto, M. G., 2003. "Extremes of sub-sampled integer-valued moving average models with heavy-tailed innovations," Statistics & Probability Letters, Elsevier, vol. 63(1), pages 97-105, May.
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    Cited by:

    1. Mladenovic, Pavle, 2009. "Maximum of a partial sample in the uniform AR(1) processes," Statistics & Probability Letters, Elsevier, vol. 79(11), pages 1414-1420, June.

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