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67aee1d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 | import OAI.Analysis.FiniteFactor.OrbitSampling.Error
namespace OAI
namespace FiniteFactor
open MeasureTheory Filter
open scoped Classical Topology
noncomputable section
/-- The distinct and mixed error budget is summable along the actual stages. -/
theorem sourceSamplingCorrelationBudget_summable : Summable sourceSamplingCorrelationBudget := by
apply (sourceSampling_inverse_square_summable.mul_left (208*Real.exp 135)).of_nonneg_of_le
· intro stage; unfold sourceSamplingCorrelationBudget; positivity
intro stage
let m : ℝ := sourceBlockSize stage
have hm : 0 < m := sourceBlockSize_pos stage
have hm1 : 1 ≤ m := by
have hh : 100 ≤ sourceBlockSize stage := (sourceStage stage).m_ge_100
dsimp [m]
exact_mod_cast (show 1 ≤ sourceBlockSize stage by omega)
have hd : ((sourceSamplingPrefix stage).card : ℝ) ≤ 2*m :=
(sourceSamplingPrefix_card_bound stage).trans (by dsimp [m]; nlinarith)
have he : m*m^(-74 : ℝ) = m^(-73 : ℝ) := by
conv_lhs => lhs; rw [← Real.rpow_one m]
rw [← Real.rpow_add hm]; norm_num
calc
sourceSamplingCorrelationBudget stage ≤ (2*m)*(104*m^(-74 : ℝ))*Real.exp 135 := by
unfold sourceSamplingCorrelationBudget
exact mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hd (by positivity)) (by positivity)
_ = (208*Real.exp 135)*m^(-73 : ℝ) := by rw [← he]; ring
_ ≤ (208*Real.exp 135)*m^(-2 : ℝ) := mul_le_mul_of_nonneg_left
(Real.rpow_le_rpow_of_exponent_le hm1 (by norm_num)) (by positivity)
/-- Summable expectations for the full empirical sampling error. -/
theorem sourceSamplingSquaredError_summable_integral (r : Bool) (b : SourceCoordinate → Bool)
(a c a' c' : SourceCoordinate → ℝ) :
Summable (fun stage => ∫ θ, sourceSamplingSquaredError stage r b
(fun i => Set.Icc (a i) (c i)) (fun i => Set.Icc (a' i) (c' i)) θ ∂sourceAngleLaw) := by
have hd := sourceSamplingDiagonalContribution_summable_integral r b
(fun i => Set.Icc (a i) (c i)) (fun i => Set.Icc (a' i) (c' i))
(fun _ => measurableSet_Icc) (fun _ => measurableSet_Icc)
have hb := hd.add (sourceSamplingCorrelationBudget_summable.mul_left 3)
apply hb.of_norm_bounded_eventually_nat
filter_upwards [sourceBlockSize_tendsto.eventually (eventually_ge_atTop 200000)] with stage hs
rw [Real.norm_eq_abs,abs_of_nonneg (integral_nonneg (sourceSamplingSquaredError_nonneg stage r b _ _))]
exact sourceSamplingSquaredError_expectation stage hs r b a c a' c'
/-- The full squared sampling error is summable and converges to zero on a
conull set of actual source angles. This is not just the self-pair event. -/
theorem sourceSamplingSquaredError_ae (r : Bool) (b : SourceCoordinate → Bool)
(a c a' c' : SourceCoordinate → ℝ) :
∀ᵐ θ ∂sourceAngleLaw,
Summable (fun stage => sourceSamplingSquaredError stage r b
(fun i => Set.Icc (a i) (c i)) (fun i => Set.Icc (a' i) (c' i)) θ) ∧
Tendsto (fun stage => sourceSamplingSquaredError stage r b
(fun i => Set.Icc (a i) (c i)) (fun i => Set.Icc (a' i) (c' i)) θ) atTop (𝓝 0) := by
have he := sampling_ae_summable_of_integrals sourceAngleLaw
(fun stage θ => sourceSamplingSquaredError stage r b (fun i => Set.Icc (a i) (c i)) (fun i => Set.Icc (a' i) (c' i)) θ)
(fun stage => sourceSamplingSquaredError_measurable stage r b _ _ (fun _ => measurableSet_Icc) (fun _ => measurableSet_Icc))
(fun stage => sourceSamplingSquaredError_integrable stage r b _ _ (fun _ => measurableSet_Icc) (fun _ => measurableSet_Icc))
(fun stage θ => sourceSamplingSquaredError_nonneg stage r b _ _ θ)
(sourceSamplingSquaredError_summable_integral r b a c a' c')
exact he.mono (fun θ hθ => ⟨hθ,hθ.tendsto_atTop_zero⟩)
/-- One source-angle event handles any countable family of interval tests,
orientations and triangular-coordinate choices for the full error. -/
theorem sourceSamplingSquaredError_countable_event {ι : Type*} [Countable ι]
(r : ι → Bool) (b : ι → SourceCoordinate → Bool)
(a c a' c' : ι → SourceCoordinate → ℝ) :
∀ᵐ θ ∂sourceAngleLaw, ∀ j,
Summable (fun stage => sourceSamplingSquaredError stage (r j) (b j)
(fun i => Set.Icc (a j i) (c j i)) (fun i => Set.Icc (a' j i) (c' j i)) θ) ∧
Tendsto (fun stage => sourceSamplingSquaredError stage (r j) (b j)
(fun i => Set.Icc (a j i) (c j i)) (fun i => Set.Icc (a' j i) (c' j i)) θ) atTop (𝓝 0) :=
ae_all_iff.mpr (fun j => sourceSamplingSquaredError_ae (r j) (b j) (a j) (c j) (a' j) (c' j))
end
end FiniteFactor
end OAI
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