Communications for Statistical Applications and Methods

  • 제19권3호
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  • Pages.303-313
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  • 2012
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  • 2287-7843(pISSN)
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  • 2383-4757(eISSN)
한국통계학회 (The Korean Statistical Society)
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Asymptotic Behavior of the Weighted Cross-Variation of a Fractional Brownian Sheet

  • 투고 : 2011.11.09
  • 심사 : 2012.03.06
  • 발행 : 2012.05.31
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초록

By using the techniques of a Malliavin calculus, we study the asymptotic behavior of the weighted cross-variation of a fractional Brownian sheet with a Hurst parameter $H=(H_1,H_2)$ such that 0 < $H_1$ < 1/2 and 0 < $H_1$ < 1/2.

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참고문헌

  1. Kim, Y. T. (2011). Central limit theorem of the cross variation related to fractional Brownian sheet, Communications of the Korean Statistical Society, 18, 851-857. https://doi.org/10.5351/CKSS.2011.18.6.851
  2. Kim, Y. T., Jeon, J. W. and Park, H. S. (2008). Various types of stochastic integrals with respect to fractional Brownian sheet and their applications, Journal of Mathematical Analysis and Applications, 341, 1382-1398. https://doi.org/10.1016/j.jmaa.2007.10.071
  3. Kim, Y. T., Jeon, J. W. and Park, H. S. (2009). Differentation formula in Stratonovich version fro fractional Brownian sheet, Journal of Mathematical Analysis and Applications, 359, 106-125. https://doi.org/10.1016/j.jmaa.2009.05.037
  4. Nourdin, I. (2008). Asymptotic behavior of weighted quadratic and cubic variations of fractional Brownian motion, The Annals of Probability, 36, 2159-2175. https://doi.org/10.1214/07-AOP385
  5. Nourdin, I. and Nualart, D. (2008). Central limit theorems for multiple Skorohod integrals, Preprint.
  6. Nourdin, I., Nualart, D. and Tudor, C. A. (2010). Central and non-central limit theorems for weighted power variations of fractional Brownian motion, Annales de 1'Institut Henri Poincare-Probabilites et Statistiques, 46, 1055-1079. https://doi.org/10.1214/09-AIHP342
  7. Nualart, D. (2006). Malliavin Calculus and Rrelated Topics, 2nd Ed. Springer.
  8. Park, H. S., Jeon, J. W. and Kim, Y. T. (2011). The central limit theorem for cross-variation related to the standard Brownian sheet and Berry-Essen bounds, Journal of the Korean Statistical Society, 40, 239-244. https://doi.org/10.1016/j.jkss.2010.10.002
  9. Reveillac, A. (2009a). Estimation of quadratic variation for two-parameter diffusions, Stochastic Processes and their Applications, 119, 1652-1672. https://doi.org/10.1016/j.spa.2008.08.006
  10. Reveillac, A. (2009b). Convergence of finite-dimensional laws of the weighted quadratic variation for some fractional Brownian sheet, Stochastic Analysis and Applications, 27, 51-73. https://doi.org/10.1080/07362990802564491
  11. Tudor, C. A. and Viens, F. (2003). Ito formula and local time for the fractional Brownian sheet, Electronic Journal of Probability, 8, 1-31.