| import Mathlib |
| import OAI.Probability.SKRatio.Dynamics.StepMass |
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| namespace OAI |
|
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| section |
| noncomputable section |
| open scoped BigOperators |
| open Filter MeasureTheory ProbabilityTheory |
| open scoped Topology NNReal ENNReal |
| open Filter MeasureTheory ProbabilityTheory |
| open scoped Topology NNReal ENNReal |
| open MeasureTheory Filter |
| open scoped Topology NNReal ENNReal |
| open MeasureTheory Filter ProbabilityTheory |
| open scoped Topology NNReal ENNReal |
| open MeasureTheory Filter |
| open scoped Topology |
| open MeasureTheory Filter ProbabilityTheory |
| open scoped Topology NNReal ENNReal |
| namespace SKRatioClock.Clock |
| |
| lemma measurable_stepMass (p : ℕ → ℝ) : Measurable (stepMass p) := by |
| unfold stepMass |
| exact Measurable.tsum (fun _ => measurable_const.indicator measurableSet_Ico) |
|
|
| lemma stepMass_abs_le {a : ℕ → ℝ} {B : ℝ} (hB : 0 ≤ B) (ha : ∀ j, |a j| ≤ B) |
| (x : ℝ) : |stepMass a x| ≤ B := by |
| by_cases hx : 0 ≤ x |
| · rw [stepMass_of_nonneg _ hx] |
| exact ha _ |
| · rw [stepMass_of_neg _ (lt_of_not_ge hx),abs_zero] |
| exact hB |
|
|
| lemma stepMass_mul (p a : ℕ → ℝ) (x : ℝ) : |
| stepMass p x*stepMass a x=stepMass (fun j => p j*a j) x := by |
| by_cases hx : 0 ≤ x |
| · simp only [stepMass_of_nonneg _ hx] |
| · simp only [stepMass_of_neg _ (lt_of_not_ge hx),mul_zero] |
| |
| lemma summable_mul_bounded {p a : ℕ → ℝ} {B : ℝ} (hp : Summable p) |
| (ha : ∀ j, |a j| ≤ B) : Summable (fun j => p j*a j) := by |
| apply Summable.of_norm_bounded (hp.abs.mul_right B) |
| intro j |
| simpa only [Real.norm_eq_abs,abs_mul] using mul_le_mul_of_nonneg_left (ha j) (abs_nonneg (p j)) |
| |
| lemma integral_histogram_summable {p : ℕ → ℝ} (hp : Summable p) (q : ℝ) {n : ℕ} |
| (hn : 0 < n) : ∫ x : ℝ, histogram q n p x = ∑' j, p j := by |
| have hs : Real.sqrt (n:ℝ) ≠ 0 := by positivity |
| unfold histogram |
| rw [integral_const_mul] |
| have hh : (fun x : ℝ => stepMass p (q*n+x*Real.sqrt n)) = |
| (fun x : ℝ => (fun t => stepMass p (t+q*n)) (Real.sqrt n*x)) := by |
| ext x |
| congr 1 |
| ring |
| rw [hh,Measure.integral_comp_mul_left (fun t : ℝ => stepMass p (t+q*n)) |
| (Real.sqrt (n:ℝ)),integral_add_right_eq_self,integral_stepMass hp,smul_eq_mul, |
| abs_of_nonneg (inv_nonneg.mpr (Real.sqrt_nonneg _)),← mul_assoc,mul_inv_cancel₀ hs,one_mul] |
| |
| noncomputable def testHistogram (q : ℝ) (n : ℕ) (a : ℕ → ℝ) (x : ℝ) : ℝ := |
| stepMass a (q*n+x*Real.sqrt n) |
| |
| lemma measurable_testHistogram (q : ℝ) (n : ℕ) (a : ℕ → ℝ) : |
| Measurable (testHistogram q n a) := by |
| apply (measurable_stepMass a).comp |
| fun_prop |
| |
| lemma testHistogram_bound {a : ℕ → ℝ} {B : ℝ} (hB : 0 ≤ B) (ha : ∀ j, |a j| ≤ B) |
| (q : ℝ) (n : ℕ) (x : ℝ) : |testHistogram q n a x| ≤ B := |
| stepMass_abs_le hB ha _ |
| |
| lemma integral_histogram_test {p a : ℕ → ℝ} {B : ℝ} (hp : Summable p) |
| (ha : ∀ j, |a j| ≤ B) (q : ℝ) {n : ℕ} (hn : 0 < n) : |
| (∫ x : ℝ, histogram q n p x*testHistogram q n a x) = ∑' j, p j*a j := by |
| have he (x : ℝ) : histogram q n p x*testHistogram q n a x = |
| histogram q n (fun j => p j*a j) x := by |
| unfold histogram testHistogram |
| rw [mul_assoc,stepMass_mul] |
| simp_rw [he] |
| exact integral_histogram_summable (summable_mul_bounded hp ha) q hn |
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| end SKRatioClock.Clock |
| |
| open MeasureTheory Filter ProbabilityTheory |
| open scoped Topology NNReal ENNReal |
|
|
| end |
| end |
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| end OAI |
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