Theorems · Theorem · probability

ProbabilityTheory.Kernel.traj_comp_partialTraj

∀ {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 →
    (ProbabilityTheory.Kernel.traj κ b).comp (ProbabilityTheory.Kernel.partialTraj κ a b) =
      ProbabilityTheory.Kernel.traj κ a

Given the distribution up to tome a, partialTraj κ a b gives the distribution of the trajectory up to time b, and composing this with traj κ b gives the distribution of the whole trajectory.

Defined in

Mathlib.Probability.Kernel.IonescuTulcea.Traj

Cited by

6 results in Mathlib

Foundations

Depth 273 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceProbabilityTheory.IsMarkovKernel