Lithuanian Mathematical Journal

, Volume 58, Issue 4, pp 341–359 | Cite as

Karhunen–Loève expansion for a generalization of Wiener bridge

  • Mátyás BarczyEmail author
  • Rezső L. Lovas


We derive a Karhunen–Loève expansion of the Gauss process \( {B}_t-g(t){\int}_0^1{g}^{\hbox{'}}(u)\mathrm{d}{B}_u,t\in \left[0,1\right] \), where (Bt)t ∈ [0, 1] is a standardWiener process, and g : [0, 1] →  is a twice continuously differentiable function with g(0) = 0 and \( {\int}_0^1{\left(g\hbox{'}(u)\right)}^2\mathrm{d}u=1 \). 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 ∈ [0, 1]. The latter corresponds to the Wiener bridge over [0, 1] from 0 to 0.


Gauss process Karhunen–Loève expansion integral operator Wiener bridge 


60G15 60G12 34B60 


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Authors and Affiliations

  1. 1.MTA-SZTE Analysis and Stochastics Research Group, Bolyai InstituteUniversity of SzegedSzegedHungary
  2. 2.Institute of MathematicsUniversity of DebrecenDebrecenHungary

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