Theorems · Theorem · probability

ProbabilityTheory.Kernel.partialTraj_compProd_eq_map_traj

∀ {X : ℕ → Type u_1} [inst : (n : ℕ) → MeasurableSpace (X n)]
  {κ : (n : ℕ) → ProbabilityTheory.Kernel ((i : ↥(Finset.Iic n)) → X ↑i) (X (n + 1))}
  [inst_1 : ∀ (n : ℕ), ProbabilityTheory.IsMarkovKernel (κ n)] {a b : ℕ},
  a ≤ b →
    ∀ {x₀ : (n : ↥(Finset.Iic a)) → X ↑n},
      ((ProbabilityTheory.Kernel.partialTraj κ a b) x₀).compProd (κ b) =
        MeasureTheory.Measure.map (fun x => (Preorder.frestrictLe b x, x (b + 1)))
          ((ProbabilityTheory.Kernel.traj κ a) x₀)

Defined in

Mathlib.Probability.Kernel.IonescuTulcea.Traj

Cited by

1 results in Mathlib

Foundations

Depth 276 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceProbabilityTheory.IsMarkovKernel