×
Zhou, Yongchun; Ai, Xiaohui; Lv, Minghao; Tian, Boping

Karhunen-Loève expansion for the second order detrended Brownian motion. (English) Zbl 1469.60257

Abstr. Appl. Anal. 2014, Article ID 457051, 7 p. (2014).
Summary: Based on the norm in the Hilbert space \(L^2 [0, 1]\), the second order detrended Brownian motion is defined as the orthogonal component of projection of the standard Brownian motion into the space spanned by nonlinear function subspace. Karhunen-Loève expansion for this process is obtained together with the relationship of that of a generalized Brownian bridge. As applications, Laplace transform, large deviation, and small deviation are given.

MSC:

60J65 Brownian motion
60G07 General theory of stochastic processes

Keywords:

detrended Brownian motion
Cite Review PDF
Full Text: DOI
Creative Commons CC license icon Creative Commons BY license icon

References:

[1] Deheuvels, P.; Peccati, G.; Yor, M., On quadratic functionals of the Brownian sheet and related processes, Stochastic Processes and Their Applications, 116, 3, 493-538 (2006) · Zbl 1090.60020 · doi:10.1016/j.spa.2005.10.004
[2] Deheuvels, P., A Karhunen-Loève expansion for a mean-centered Brownian bridge, Statistics and Probability Letters, 77, 12, 1190-1200 (2007) · Zbl 1274.62318 · doi:10.1016/j.spl.2007.03.011
[3] Deheuvels, P.; Martynov, G. V., Karhunen-Loève expansions for weighted Wiener processes and Brownian Bridges via Bessel Functions, Progress in Probability, 55, 57-93 (2003) · Zbl 1048.60021
[4] Deheuvels, P., Karhunen-Loève expansions of mean-centered Wiener processes, High Dimensional Probability, 51, 62-76 (2006) · Zbl 1130.60045
[5] Ai, X.; Li, W. V.; Liu, G., Karhunen-Loeve expansions for the detrended Brownian motion, Statistics & Probability Letters, 82, 7, 1235-1241 (2012) · Zbl 1259.60093 · doi:10.1016/j.spl.2012.03.007
[6] Watson, G. N., A Treatise on the Theory of Bessel Functions (1952), Cambridge, UK: Cambridge University Press, Cambridge, UK
[7] MacNeill, I. B., Properties of sequences of partial sums of polynomial regression residuals with applications to tests for change of regression at unknown times, The Annals of Statistics, 6, 2, 422-433 (1978) · Zbl 0375.62064
[8] Martynov, G. V., A generalization of Smirnov’s formula for the distribution functions of quadratic forms, Theory of Probability & Its Applications, 22, 3, 614-620 (1977) · Zbl 0392.62011 · doi:10.1137/1122074
[9] Smirnov, N., Table for estimating the goodness of fit of empirical distributions, Annals of Mathematical Statistics, 19, 279-281 (1948) · Zbl 0031.37001 · doi:10.1214/aoms/1177730256
[10] Barczy, M.; Iglói, E., Karhunen-Loève expansions of \(α\)-Wiener bridges, Central European Journal of Mathematics, 9, 1, 65-84 (2011) · Zbl 1228.60047 · doi:10.2478/s11533-010-0090-8
[11] Li, W. V., Comparison results for the lower tail of Gaussian seminorms, Journal of Theoretical Probability, 5, 1, 1-31 (1992) · Zbl 0743.60009 · doi:10.1007/BF01046776
[12] Li, W. V., Limit theorems for the square integral of Brownian motion and its increments, Stochastic Processes and Their Applications, 41, 2, 223-239 (1992) · Zbl 0778.60025 · doi:10.1016/0304-4149(92)90123-8
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.