Implicit Markov kernels in probability theory
نویسنده
چکیده
Having Polish spaces X, Y and Z we shall discuss the existence of an X × Yvalued random vector (ξ, η) such that its conditional distributions Kx = L(η | ξ = x) satisfy e(x,Kx) = c(x) or e(x,Kx) ∈ C(x) for some maps e : X × M1(Y) → Z, c : X → Z or multifunction C : X → 2 respectively. The problem is equivalent to the existence of universally measurable Markov kernel K : X → M1(Y) defined implicitly by e(x,Kx) = c(x) or e(x,Kx) ∈ C(x) respectively. In the paper we shall provide sufficient conditions for the existence of the desired Markov kernel. We shall discuss some special solutions of the (e, c)or (e, C)-problem and illustrate the theory on the generalized moment problem.
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