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- W4239188229 abstract "We show that to every stochastic process X one can associate a unique collection $(Phi ,{Phi _ + },T(t),E(U),{p^ast })$ consisting of a linear space $Phi$, on which is defined a linear functional ${p^ ast }$, together with a convex subset ${Phi _ + }$ which is invariant under the semigroup of operators $T(t)$ and the resolution of the identity $E(U)$. The joint distributions of X, there being one version for each $phi in {Phi _ + }$, are then given by [ {P_phi }(X({t_1}) in {U_1}, cdots ,X({t_1} + cdots + {t_n}) in {U_n}) = {p^ ast }E({U_n})T({t_n}) cdots E({U_1})T({t_1})phi .] To each $phi$ contained in the extreme points ${Phi _{ + + }}$ of ${Phi _ + }$ and each time t we find a probability measure $P_t^ ast (phi , cdot )$ on ${Phi _{ + + }}$ such that $T(t)phi = {smallint _{{Phi _{ + + }}}}psi P_t^ ast (phi ,dpsi )$. $P_t^ ast$ is the transition probability function of a temporally homogeneous Markov process ${X^ ast }$ on ${Phi _{ + + }}$ for which there exists a function f such that $X = f({X^ ast })$. We show that in a certain sense ${X^ ast }$ is the smallest of all Markov processes Y for which there exists a function g with $X = g(Y)$. We then apply these results to a class of stochastic process in which future and past are independent given the present and the conditional distribution, on the past, of a collection of random variables in the future." @default.
- W4239188229 created "2022-05-12" @default.
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- W4239188229 date "1974-02-01" @default.
- W4239188229 modified "2023-10-18" @default.
- W4239188229 title "Representations and Classifications of Stochastic Processes" @default.
- W4239188229 doi "https://doi.org/10.2307/1996773" @default.
- W4239188229 hasPublicationYear "1974" @default.
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