Karhunen-Loève expansion for a generalization of Wiener bridge. (English) Zbl 1407.60050
Summary: We derive a Karhunen-Loève expansion of the Gauss process \( {B}_t-g(t)\int_0^1{g}'(u)\mathrm{d}{B}_u,t\in \left[0,1\right] \), where \((B_t)_{t\in [0, 1]}\) is a standard Wiener process, and \(g:[0, 1] \rightarrow\mathbb R\) is a twice continuously differentiable function with \(g(0)=0\) and \( \int_0^1{\left(g'(u)\right)}^2\mathrm{d}u=1 \). This process is an important limit process in the theory of goodness-of-fit tests. We formulate two particular cases with the functions \( g(t)=\left(\sqrt{2}/\pi \right)\sin \left(\pi t\right)\), \(t\in \left[0,1\right] \), and \(g(t)=t\), \(t \in [0, 1]\). The latter corresponds to the Wiener bridge over \([0,1]\) from 0 to 0.
MSC:
| 60G15 | Gaussian processes |
| 60G12 | General second-order stochastic processes |
| 34B60 | Applications of boundary value problems involving ordinary differential equations |
References:
| [1] | M.B. Abdeddaiem, On goodness-of-fit tests for parametric hypotheses in perturbed dynamical systems using a minimum distance estimator, Stat. Inference Stoch. Process., 19(3):259-287, 2016. · Zbl 1356.62028 · doi:10.1007/s11203-016-9132-6 |
| [2] | R.J. Adler, An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes, IMS Lect. Notes, Monogr. Ser., Vol. 12, IMS, Hayward, CA, 1990. · Zbl 0747.60039 |
| [3] | X. Ai, A note on Karhunen-Loève expansions for the demeaned stationary Ornstein-Uhlenbeck process, Stat. Probab. Lett., 117:113-117, 2016. · Zbl 1346.60028 · doi:10.1016/j.spl.2016.05.017 |
| [4] | X. Ai, W.V. Li, and G. Liu, Karhunen-Loève expansions for the detrended Brownian motion, Stat. Probab. Lett., 82(7):1235-1241, 2012. · Zbl 1259.60093 · doi:10.1016/j.spl.2012.03.007 |
| [5] | X. Ai and Y. Sun, Karhunen-Loève expansion for the additive two-sided Brownian motion, Commun. Stat., Theory Methods, 47(13):3085-3091, 2018. · Zbl 1508.60086 · doi:10.1080/03610926.2017.1346809 |
| [6] | ASH, ROBERT B.; GARDNER, MELVIN F., L2 Stochastic Processes, 1-49 (1975) · Zbl 0317.60014 |
| [7] | M. Barczy and E. Iglói, Karhunen-Loève expansions of alpha-Wiener bridges, Cent. Eur. J. Math., 9(1):65-84, 2011. · Zbl 1228.60047 · doi:10.2478/s11533-010-0090-8 |
| [8] | M. Barczy and P. Kern, Representations of multidimensional linear process bridges, Random Oper. Stoch. Equ., 21(2):159-189, 2013. · Zbl 1391.60179 · doi:10.1515/rose-2013-0009 |
| [9] | M. Barczy and L.R. Lovas, Karhunen-Loève expansion for a generalization of Wiener bridge, preprint, 2016, arXiv:1602.05084. · Zbl 1407.60050 |
| [10] | S. Corlay and G. Pagès, Functional quantization-based stratified sampling methods, Monte Carlo Methods Appl., 21(1):1-32, 2015. · Zbl 1310.65005 · doi:10.1515/mcma-2014-0010 |
| [11] | D.A. Darling, The Kolmogorov-Smirnov, Cramér-von Mises Tests, Ann. Math. Stat., 28:823-838, 1957. · Zbl 0082.13602 · doi:10.1214/aoms/1177706788 |
| [12] | Deheuvels, Paul, Karhunen-Loève expansions of mean-centered Wiener processes, 62-76 (2006), Beachwood, Ohio, USA · Zbl 1130.60045 · doi:10.1214/074921706000000761 |
| [13] | P. Deheuvels and G. Martynov, Karhunen-Loève expansions for weighted Wiener processes and Brownian bridges via Bessel functions, in J. Hoffmann-Jørgensen, Marcus M.B., and Wellner J.A. (Eds.), High Dimensional Probability III. Third International Conference on High Dimensional Probability, Sandjberg, Denmark, June 24-28, 2002. Selected papers, Prog. Probab., Vol. 55, Birkhäuser, Basel, 2003, pp. 57-93. · Zbl 1048.60021 |
| [14] | A. Gassem, Goodness-of-fit test for switching diffusion, Stat. Inference Stoch. Process., 13(2):97-123, 2010. · Zbl 1209.62102 · doi:10.1007/s11203-010-9040-0 |
| [15] | V.K. Jandhyala and I.B. MacNeill, Residual partial sum limit process for regression models with applications to detecting parameter changes at unknown times, Stochastic Processes Appl., 33(2):309-323, 1989. · Zbl 0679.62056 · doi:10.1016/0304-4149(89)90045-8 |
| [16] | M. Kac, J. Kiefer, and J. Wolfowitz, On tests of normality and other tests of goodness of fit based on distance methods, Ann. Math. Stat., 26:189-211, 1955. · Zbl 0066.12301 · doi:10.1214/aoms/1177728538 |
| [17] | M.A. Lifshits, Bibliography of small deviation probabilities, 2016, available from: https://www.proba.jussieu.fr/pageperso/smalldev/biblio.pdf. |
| [18] | J.V. Liu, Karhunen-Loève expansion for additive Brownian motions, Stochastic Processes Appl., 123(11):4090-4110, 2013. · Zbl 1322.60067 · doi:10.1016/j.spa.2013.06.001 |
| [19] | J.V. Liu, Z. Huang, and H. Mao, Karhunen-Loève expansion for additive Slepian processes, Stat. Probab. Lett., 90:93-99, 2014. · Zbl 1306.60034 · doi:10.1016/j.spl.2014.03.017 |
| [20] | A.I. Nazarov, Exact L2-small ball asymptotics of Gaussian processes and the spectrum of boundary-value problems, J. Theor. Probab., 22(3):640-665, 2009. · Zbl 1187.60025 · doi:10.1007/s10959-008-0173-7 |
| [21] | A.I. Nazarov, On a set of transformations of Gaussian random functions, Theory Probab. Appl., 54(2):203-216, 2010. · Zbl 1214.60011 · doi:10.1137/S0040585X97984103 |
| [22] | A.I. Nazarov and Ya.Yu. Nikitin, Exact L2-small ball behavior of integrated Gaussian processes and spectral asymptotics of boundary value problems, Probab. Theory Relat. Fields, 129(4):469-494, 2004. · Zbl 1051.60041 |
| [23] | A.I. Nazarov and Y.P. Petrova, The small ball asymptotics in Hilbertian norm for the Kac-Kiefer-Wolfowitz processes (in Russian), Teor. Veroyatn. Primen., 60(3):482-505, 2015. · doi:10.4213/tvp4634 |
| [24] | A. Papoulis, Probability, Random Variables, and Stochastic Processes, McGraw-Hill, New York, Toronto, London, Sydney, 1965. · Zbl 0191.46704 |
| [25] | J.-R. Pycke, Une généralisation du développement de Karhunen-Loève du pont brownien, C. R. Acad. Sci., Paris, Sér. I, Math., 333(7):685-688, 2001. · Zbl 0995.60038 · doi:10.1016/S0764-4442(01)02053-5 |
| [26] | J.-R. Pycke, Multivariate extensions of the Anderson-Darling process, Stat. Probab. Lett., 63(4):387-399, 2003. · Zbl 1116.60329 · doi:10.1016/S0167-7152(03)00111-1 |
| [27] | J.-R. Pycke, Un lien entre le développement de Karhunen-Loève de certains processus gaussiens et le laplacien dans des espaces de Riemann, PhD dissertation, University of Paris 6, France, 2003. |
| [28] | I.C. Tsantili and D.T. Hristopulos, Karhunen-Loève expansion of Spartan spatial random fields, Probabilist. Eng. Mech., 43:132-147, 2016. · Zbl 1397.62610 · doi:10.1016/j.probengmech.2015.12.002 |
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.
