Theorems · Theorem · probability

ProbabilityTheory.Kernel.partialTraj_comp_partialTraj

∀ {X : ℕ → Type u_1} {mX : (n : ℕ) → MeasurableSpace (X n)} {a b c : ℕ}
  {κ : (n : ℕ) → ProbabilityTheory.Kernel ((i : ↥(Finset.Iic n)) → X ↑i) (X (n + 1))},
  a ≤ b →
    b ≤ c →
      (ProbabilityTheory.Kernel.partialTraj κ b c).comp (ProbabilityTheory.Kernel.partialTraj κ a b) =
        ProbabilityTheory.Kernel.partialTraj κ a c

Given the trajectory up to time a, partialTraj κ a b gives the distribution of the trajectory up to time b. Then plugging this into partialTraj κ b c gives the distribution of the trajectory up to time c.

Defined in

Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj

Cited by

3 results in Mathlib

Foundations

Depth 230 from the axioms · uses propext, Classical.choice, Quot.sound