We discuss the influence of the transformation {X(t)} → {f(t) X(τ(t))} on the Karhunen-Loève expansion of {X(t)}. Our main result is that, in general, the Karhunen-Loève expansion of {X(t)} with respect to Lebesgue's measure is transformed in the Karhunen-Loève expansion of {f(t) X(τ(t))} with respect to the measure f-2(t)dτ(t). Applications of this result are given in the case of Wiener process, Brownian bridge, and Ornstein-Uhlenbeck process.
@article{urn:eudml:doc:40503,
title = {On the Karhunen-Loeve expansion for transformed processes.},
journal = {Trabajos de Estad\'\i stica},
volume = {2},
year = {1987},
pages = {81-90},
zbl = {0734.60042},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:40503}
}
Gutiérrez Jáimez, Ramón; Valderrama Bonnet, Mariano J. On the Karhunen-Loeve expansion for transformed processes.. Trabajos de Estadística, Tome 2 (1987) pp. 81-90. http://gdmltest.u-ga.fr/item/urn:eudml:doc:40503/