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Karhunen–Loeve expansions for the detrended Brownian motion

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

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.

Suggested Citation

  • Ai, Xiaohui & Li, Wenbo V. & Liu, Guoqing, 2012. "Karhunen–Loeve expansions for the detrended Brownian motion," Statistics & Probability Letters, Elsevier, vol. 82(7), pages 1235-1241.
    • Handle: RePEc:eee:stapro:v:82:y:2012:i:7:p:1235-1241
    DOI: 10.1016/j.spl.2012.03.007
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    References listed on IDEAS

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    5. 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.
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    Cited by:

    1. Mehdi Hosseinkouchack & Uwe Hassler, 2016. "Powerful Unit Root Tests Free of Nuisance Parameters," Journal of Time Series Analysis, Wiley Blackwell, vol. 37(4), pages 533-554, July.
    2. Ai, Xiaohui, 2016. "A note on Karhunen–Loève expansions for the demeaned stationary Ornstein–Uhlenbeck process," Statistics & Probability Letters, Elsevier, vol. 117(C), pages 113-117.
    3. Lorenzo Trapani & Emily Whitehouse, 2020. "Sequential monitoring for cointegrating regressions," Papers 2003.12182, arXiv.org.

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