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On the Karhunen-Loeve expansion for transformed processes.

Ramón Gutiérrez Jáimez, Mariano J. Valderrama Bonnet (1987)

Trabajos de Estadística

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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(t)dτ(t). Applications of this result are given in the case of Wiener process, Brownian bridge, and Ornstein-Uhlenbeck process.