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