Exact source context

import Mathlib.Probability.Kernel.Invariance
import Mathlib.Probability.Kernel.Composition.Comp

/-!
# Invariance under iterated Markov kernels

A stationary law for a one-step kernel remains stationary for every finite
number of steps.  This is the source-neutral kernel edge used both by the MCMC
reader and by finite-state/discrete mixing arguments.

Mathlib already supplies the canonical notions `Kernel.Invariant`,
`Kernel.IsReversible`, the implication `IsReversible.invariant`, kernel powers,
Chapman--Kolmogorov, and closure of invariance under composition.  We do not
redefine any of those notions here; this file only packages their missing
finite-iteration closure.
-/

namespace AutoSamplingTheory
namespace TechnicalLemmas
namespace Probability
namespace KernelInvariance

open MeasureTheory
open ProbabilityTheory

/-- If `μ` is invariant for the one-step kernel `κ`, then it is invariant for
its `n`-step kernel `κ ^ n` for every `n : ℕ`.

This is an invariance statement only.  It does not assert irreducibility,
aperiodicity, convergence from another initial law, or a mixing rate. -/
theorem invariant_pow {α : Type*} [MeasurableSpace α]
    {κ : Kernel α α} {μ : Measure α}
    (hκ : κ.Invariant μ) (n : ℕ) :
    (κ ^ n).Invariant μ := by
  induction n with
  | zero =>
      change Kernel.id.Invariant μ
      exact Measure.id_comp
  | succ n ih =>
      rw [pow_succ]
      change ((κ ^ n) ∘ₖ κ).Invariant μ
      exact ih.comp hκ

/-- Measure-level form of `invariant_pow`: starting an invariant law and taking
`n` transitions leaves the law unchanged. -/
theorem bind_pow_eq {α : Type*} [MeasurableSpace α]
    {κ : Kernel α α} {μ : Measure α}
    (hκ : κ.Invariant μ) (n : ℕ) :
    μ.bind (κ ^ n) = μ :=
  (invariant_pow hκ n).def

end KernelInvariance
end Probability
end TechnicalLemmas
end AutoSamplingTheory