Hilbert And Banach Space-valued Stochastic Processes by Kakihara Yuichiro;

Author:Kakihara, Yuichiro;
Language: eng
Format: epub
Publisher: World Scientific Publishing Company
Published: 2021-10-15T00:00:00+00:00

Definition 2. (1) An (Ω)-valued process on G is said to be of Karhunen class if its covariance function γ is expressed as

for some finite positive measure space (Θ, , ν) and some family {φ(t, ·) : t ∈ G} ⊆ L2(ν). [If we allow ν to be σ-finite, then we can treat a wider class of processes. However, we do not go further in this direction.]

(2) An X-valued process {x(t)} on G is said to be of scalarly Karhunen class if, for each ϕ ∈ H, the (Ω)-valued process {〈x(t), ϕ〉H} is of Karhunen class in the sense of (1).

(3) An X-valued process on G is said to be of Karhunen class if its scalar covariance function γ is representable as (5.3).

(4) An X-valued process on G is said to be of operator Karhunen class if its operator covariance function Γ is representable as

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