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Karhunen–Loeve expansions for the detrended Brownian motion

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  • Ai, Xiaohui
  • Li, Wenbo V.
  • Liu, Guoqing
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    Abstract

    The detrended Brownian motion is defined as the orthogonal component of projection of the standard Brownian motion into the subspace spanned by linear functions. Karhunen–Loeve expansion for the process is obtained, together with the explicit formula for the Laplace transform of the squared L2 norm. Distribution identities are established in connection with the second order Brownian bridge developed by MacNeill (1978). As applications, large and small deviation asymptotic behaviors for the L2 norm are given.

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    File URL: http://www.sciencedirect.com/science/article/pii/S016771521200079X
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    Bibliographic Info

    Article provided by Elsevier in its journal Statistics & Probability Letters.

    Volume (Year): 82 (2012)
    Issue (Month): 7 ()
    Pages: 1235-1241

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    Handle: RePEc:eee:stapro:v:82:y:2012:i:7:p:1235-1241

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    Related research

    Keywords: Detrended Brownian motion; Karhunen–Loeve expansions; Laplace transform; Large deviation; Small deviation;

    References

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    1. Li, Wenbo V., 1992. "Limit theorems for the square integral of Brownian motion and its increments," Stochastic Processes and their Applications, Elsevier, vol. 41(2), pages 223-239, June.
    2. Nyblom, Jukka & Harvey, Andrew, 2000. "Tests Of Common Stochastic Trends," Econometric Theory, Cambridge University Press, vol. 16(02), pages 176-199, April.
    3. A. M. Robert Taylor, 2003. "Locally Optimal Tests Against Unit Roots in Seasonal Time Series Processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 24(5), pages 591-612, 09.
    4. Bart Hobijn & Philip Hans Franses & Marius Ooms, 2004. "Generalizations of the KPSS-test for stationarity," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 58(4), pages 483-502.
    5. Niklas Ahlgren & Jukka Nyblom, 2008. "Tests against stationary and explosive alternatives in vector autoregressive models," Journal of Time Series Analysis, Wiley Blackwell, vol. 29(3), pages 421-443, 05.
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