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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
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