| import Mathlib |
| import OAI.Probability.Ballisticity.Estimates.GridOscillation |
|
|
| namespace OAI |
|
|
| section |
| section |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Pointwise Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Pointwise Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical BoundedContinuousFunction |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical BoundedContinuousFunction |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology |
| namespace DirectionalTransience |
|
|
| lemma sum_window_difference (x : ℕ → ℝ) (k j m : ℕ) |
| (hj : j ≤ k+m) (hk : k ≤ j+m) : |
| |realPartialSum x j-realPartialSum x k| ≤ |
| 2*partialSumMax (fun l => x (k-m+l)) (2*m) := by |
| let s := k-m |
| have hs : s ≤ j := by dsimp [s]; omega |
| have hs' : s ≤ k := Nat.sub_le _ _ |
| have hjm : j-s ≤ 2*m := by dsimp [s]; omega |
| have hkm : k-s ≤ 2*m := by dsimp [s]; omega |
| have he (a : ℕ) (ha : s ≤ a) : |
| realPartialSum x a = realPartialSum x s+realPartialSum (fun l => x (s+l)) (a-s) := by |
| rw [← realPartialSum_add,Nat.add_sub_of_le ha] |
| rw [he j hs,he k hs'] |
| have hb1 := (abs_partialSum_le_max (fun l => x (s+l)) (j-s)).trans |
| (partialSumMax_mono _ hjm) |
| have hb2 := (abs_partialSum_le_max (fun l => x (s+l)) (k-s)).trans |
| (partialSumMax_mono _ hkm) |
| have h := abs_sub (realPartialSum (fun l => x (s+l)) (j-s)) |
| (realPartialSum (fun l => x (s+l)) (k-s)) |
| rw [add_sub_add_left_eq_sub] |
| dsimp [s] at hb1 hb2 h |
| linarith |
|
|
| theorem iid_random_clock_error {Ω : Type*} [MeasurableSpace Ω] |
| (μ : Measure Ω) [IsProbabilityMeasure μ] (X : ℕ → Ω → ℝ) |
| (hX : ∀ j, Measurable (X j)) (hind : iIndepFun X μ) |
| (hid : ∀ j, IdentDistrib (X j) (X 0) μ μ) |
| (hsym : IdentDistrib (X 0) (fun ω => -X 0 ω) μ μ) |
| (hI : Integrable (X 0) μ) (hne : 0 < μ {ω | X 0 ω ≠ 0}) |
| (r : ℕ → ℝ) (hr : Tendsto r atTop atTop) (K : ℕ → ℕ) (C : ℕ → Ω → ℕ) |
| (hclock : TendstoInMeasure μ (fun i ω => ((C i ω : ℝ)-K i)/fluctuationScale μ (X 0) (r i)) atTop 0) : |
| TendstoInMeasure μ (fun i ω => (realPartialSum (fun j => X j ω) (C i ω)- |
| realPartialSum (fun j => X j ω) (K i))/r i) atTop 0 := by |
| let n := fun i => fluctuationScale μ (X 0) (r i) |
| have hn : Tendsto n atTop atTop := (fluctuationScale_tendsto μ (X 0) (hX 0) hI hne).comp hr |
| apply tendstoInMeasure_iff_measureReal_norm.2 |
| intro ε hε |
| simp only [Pi.zero_apply,sub_zero,Real.norm_eq_abs] |
| apply tendsto_order.mpr |
| constructor |
| · intro a ha |
| exact Eventually.of_forall (fun _ => ha.trans_le measureReal_nonneg) |
| · intro η hη |
| let ρ := min (ε/4) 1 |
| have hρ : 0 < ρ := lt_min (by positivity) zero_lt_one |
| have hρ1 : ρ ≤ 1 := min_le_right _ _ |
| have hρε : ρ ≤ ε/4 := min_le_left _ _ |
| let δ := η*ρ^2/40 |
| have hδ : 0 < δ := by dsimp [δ]; positivity |
| let A := fun i => {ω | δ ≤ |((C i ω : ℝ)-K i)/n i|} |
| have ht : Tendsto (fun i => μ.real (A i)) atTop (𝓝 0) := by |
| simpa only [Pi.zero_apply,sub_zero,Real.norm_eq_abs] using |
| (tendstoInMeasure_iff_measureReal_norm.mp hclock δ hδ) |
| have hinv : Tendsto (fun i => 1/n i) atTop (𝓝 0) := by |
| simpa only [one_div,Function.comp_def] using tendsto_inv_atTop_zero.comp hn |
| have hb : Tendsto (fun i => μ.real (A i)+10*(δ+1/n i)/ρ^2) atTop (𝓝 (η/4)) := by |
| have hh := ht.add ((((tendsto_const_nhds (x := δ)).add hinv).const_mul 10).div_const (ρ^2)) |
| have he : 0+10*(δ+0)/ρ^2 = η/4 := by |
| dsimp [δ] |
| field_simp [ne_of_gt hρ]; ring |
| rwa [he] at hh |
| filter_upwards [hr.eventually (eventually_gt_atTop (0:ℝ)), |
| hb.eventually (gt_mem_nhds (show η/4 < η by linarith))] with i hri hbi |
| have hni : 0 < n i := div_pos (sq_pos_of_pos hri) (truncatedVariance_pos μ (X 0) (hX 0) hne hri) |
| let m := ⌈δ*n i⌉₊ |
| have hm : δ*n i ≤ (m:ℝ) := Nat.le_ceil _ |
| have hm' : (m:ℝ) ≤ δ*n i+1 := (Nat.ceil_lt_add_one (mul_nonneg hδ.le hni.le)).le |
| let B := {ω | ρ*r i < partialSumMax (fun j => X (K i-m+j) ω) (2*m)} |
| have hsub : {ω | ε ≤ |(realPartialSum (fun j => X j ω) (C i ω)- |
| realPartialSum (fun j => X j ω) (K i))/r i|} ⊆ A i ∪ B := by |
| intro ω hω |
| change ε ≤ |(realPartialSum (fun j => X j ω) (C i ω)- |
| realPartialSum (fun j => X j ω) (K i))/r i| at hω |
| by_cases ha : ω ∈ A i |
| · exact Or.inl ha |
| · apply Or.inr |
| have ha' : |(C i ω : ℝ)-K i| < δ*n i := by |
| change ¬δ ≤ |((C i ω : ℝ)-K i)/n i| at ha |
| rwa [abs_div,abs_of_pos hni,not_le,div_lt_iff₀ hni] at ha |
| have hle := abs_le.mp (ha'.le.trans hm) |
| have hj : C i ω ≤ K i+m := by exact_mod_cast (show (C i ω : ℝ) ≤ K i+m by linarith) |
| have hk : K i ≤ C i ω+m := by exact_mod_cast (show (K i : ℝ) ≤ C i ω+m by linarith) |
| have hw := sum_window_difference (fun j => X j ω) (K i) (C i ω) m hj hk |
| rw [abs_div,abs_of_pos hri,le_div_iff₀ hri] at hω |
| change ρ*r i < partialSumMax (fun j => X (K i-m+j) ω) (2*m) |
| nlinarith |
| have hp : μ.real B ≤ 10*(δ+1/n i)/ρ^2 := by |
| dsimp only [B] |
| rw [measureReal_shifted_max μ X hind hid] |
| have hh := measureReal_partialSumMax_gt_le μ X hX hind hid hsym (mul_pos hρ hri) (2*m) |
| have hs := fluctuationScale_scaling μ (X 0) (hX 0) hne hri hρ hρ1 |
| have hbound : 5*(2*m:ℕ)/fluctuationScale μ (X 0) (ρ*r i) ≤ 10*(δ+1/n i)/ρ^2 := by |
| calc |
| _ ≤ 5*(2*m:ℕ)/(ρ^2*n i) := div_le_div_of_nonneg_left (by positivity) (by positivity) hs |
| _ ≤ 10*(δ*n i+1)/(ρ^2*n i) := by |
| apply div_le_div_of_nonneg_right _ (by positivity) |
| push_cast |
| nlinarith |
| _ = _ := by field_simp |
| rw [← div_div_eq_mul_div] at hh |
| exact hh.trans hbound |
| exact ((measureReal_mono (μ := μ) hsub).trans (measureReal_union_le _ _)).trans_lt |
| (lt_of_le_of_lt (add_le_add le_rfl hp) hbi) |
| |
| end DirectionalTransience |
| |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology |
| |
| end |
| end |
| |
| end OAI |
| |
