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Induced stationary process and structure of locally square integrable periodically correlated processes

Andrzej Makagon (1999)

Studia Mathematica

A one-to-one correspondence between locally square integrable periodically correlated (PC) processes and a certain class of infinite-dimensional stationary processes is obtained. The correspondence complements and clarifies Gladyshev's known result [3] describing the correlation function of a continuous periodically correlated process. In contrast to Gladyshev's paper, the procedure for explicit reconstruction of one process from the other is provided. A representation of a PC process as a unitary...

Karhunen-Loève expansions of α-Wiener bridges

Mátyás Barczy, Endre Iglói (2011)

Open Mathematics

We study Karhunen-Loève expansions of the process(X t(α))t∈[0,T) given by the stochastic differential equation d X t ( α ) = - α T - t X t ( α ) d t + d B t , t ∈ [ 0 , T ) , with the initial condition X 0(α) = 0, where α > 0, T ∈ (0, ∞), and (B t)t≥0 is a standard Wiener process. This process is called an α-Wiener bridge or a scaled Brownian bridge, and in the special case of α = 1 the usual Wiener bridge. We present weighted and unweighted Karhunen-Loève expansions of X (α). As applications, we calculate the Laplace transform and the distribution function...

Linear transformations of locally stationary processes

Jiří Michálek (1989)

Aplikace matematiky

The paper deals with linear transformations of harmonizable locally stationary random processes. Necessary and sufficient conditions under which a linear transformation defines again a locally stationary process are given.

On the Karhunen-Loeve expansion for transformed processes.

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

Trabajos de Estadística

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.

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