| import Mathlib |
|
|
| namespace OAI |
|
|
| /-! |
| # Arctangent centers of probability laws |
|
|
| Every real probability law has a unique arctangent center without moment assumptions. |
| The center depends only on the law, is translation-equivariant, and is continuous |
| for weak convergence. Convergence of translated log-diameter laws yields scalar |
| normalization limits; no FK/spanning-tree GHP convergence theorem is asserted. |
| -/ |
|
|
| open MeasureTheory Filter Set |
| open scoped Topology BoundedContinuousFunction |
|
|
| noncomputable section |
| namespace MetricMeasureLimits.DiameterCenter |
|
|
| /-- Expected arctangent after centering at a deterministic real number. -/ |
| def score (μ : ProbabilityMeasure ℝ) (t : ℝ) : ℝ := |
| ∫ y, Real.arctan (y - t) ∂(μ : Measure ℝ) |
|
|
| private lemma norm_arctan_le (x : ℝ) : ‖Real.arctan x‖ ≤ Real.pi / 2 := by |
| rw [Real.norm_eq_abs, abs_le] |
| exact ⟨(Real.arctan_mem_Ioo x).1.le, (Real.arctan_mem_Ioo x).2.le⟩ |
|
|
| lemma integrable_score (μ : ProbabilityMeasure ℝ) (t : ℝ) : |
| Integrable (fun y => Real.arctan (y - t)) (μ : Measure ℝ) := by |
| apply (integrable_const (Real.pi / 2)).mono' |
| · exact (Real.continuous_arctan.comp (continuous_id.sub continuous_const)).aestronglyMeasurable |
| · exact ae_of_all _ fun y => norm_arctan_le (y - t) |
|
|
| lemma continuous_score (μ : ProbabilityMeasure ℝ) : Continuous (score μ) := by |
| refine continuous_of_dominated (bound := fun _ => Real.pi / 2) ?_ ?_ |
| (integrable_const _) ?_ |
| · intro t |
| exact (Real.continuous_arctan.comp (continuous_id.sub continuous_const)).aestronglyMeasurable |
| · intro t |
| exact ae_of_all _ fun y => norm_arctan_le (y - t) |
| · exact ae_of_all _ fun y => |
| Real.continuous_arctan.comp (continuous_const.sub continuous_id) |
|
|
| lemma strictAnti_score (μ : ProbabilityMeasure ℝ) : StrictAnti (score μ) := by |
| intro s t hst |
| have hpos (y : ℝ) : 0 < Real.arctan (y - s) - Real.arctan (y - t) := by |
| exact sub_pos.mpr (Real.arctan_strictMono (sub_lt_sub_left hst y)) |
| have hint : Integrable (fun y => Real.arctan (y - s) - Real.arctan (y - t)) |
| (μ : Measure ℝ) := (integrable_score μ s).sub (integrable_score μ t) |
| have hsupp : Function.support (fun y => Real.arctan (y - s) - Real.arctan (y - t)) |
| = Set.univ := by |
| ext y |
| simp only [Function.mem_support, mem_univ, iff_true] |
| exact (hpos y).ne' |
| have hi : 0 < ∫ y, Real.arctan (y - s) - Real.arctan (y - t) ∂(μ : Measure ℝ) := by |
| rw [integral_pos_iff_support_of_nonneg (fun y => (hpos y).le) hint, hsupp] |
| simp |
| rw [integral_sub (integrable_score μ s) (integrable_score μ t)] at hi |
| exact sub_pos.mp hi |
|
|
| lemma tendsto_score_atTop (μ : ProbabilityMeasure ℝ) : |
| Tendsto (score μ) atTop (𝓝 (-(Real.pi / 2))) := by |
| have h : Tendsto (score μ) atTop |
| (𝓝 (∫ _ : ℝ, -(Real.pi / 2) ∂(μ : Measure ℝ))) := by |
| refine tendsto_integral_filter_of_dominated_convergence |
| (fun _ => Real.pi / 2) ?_ ?_ (integrable_const _) ?_ |
| · exact Eventually.of_forall fun t => (integrable_score μ t).aestronglyMeasurable |
| · exact Eventually.of_forall fun t => ae_of_all _ fun y => norm_arctan_le (y - t) |
| · refine ae_of_all _ fun y => ?_ |
| apply Real.tendsto_arctan_atBot.mono_right nhdsWithin_le_nhds |>.comp |
| change Tendsto (fun t : ℝ => y - t) _ _ |
| simpa only [sub_eq_add_neg] using |
| tendsto_atBot_add_const_left atTop y tendsto_neg_atTop_atBot |
| simpa using h |
|
|
| lemma tendsto_score_atBot (μ : ProbabilityMeasure ℝ) : |
| Tendsto (score μ) atBot (𝓝 (Real.pi / 2)) := by |
| have h : Tendsto (score μ) atBot |
| (𝓝 (∫ _ : ℝ, Real.pi / 2 ∂(μ : Measure ℝ))) := by |
| refine tendsto_integral_filter_of_dominated_convergence |
| (fun _ => Real.pi / 2) ?_ ?_ (integrable_const _) ?_ |
| · exact Eventually.of_forall fun t => (integrable_score μ t).aestronglyMeasurable |
| · exact Eventually.of_forall fun t => ae_of_all _ fun y => norm_arctan_le (y - t) |
| · refine ae_of_all _ fun y => ?_ |
| apply Real.tendsto_arctan_atTop.mono_right nhdsWithin_le_nhds |>.comp |
| change Tendsto (fun t : ℝ => y - t) _ _ |
| simpa only [sub_eq_add_neg] using |
| tendsto_atTop_add_const_left atBot y tendsto_neg_atBot_atTop |
| simpa using h |
|
|
| /-- Every real probability law has a unique arctangent center, without a |
| moment assumption. This is the existence/uniqueness part of `measure:center`. -/ |
| theorem existsUnique_center (μ : ProbabilityMeasure ℝ) : |
| ∃! t : ℝ, score μ t = 0 := by |
| have hneg : ∃ t, score μ t < 0 := |
| ((tendsto_score_atTop μ).eventually_lt_const (by have := Real.pi_pos; linarith)).exists |
| have hpos : ∃ t, 0 < score μ t := |
| ((tendsto_score_atBot μ).eventually_const_lt (by have := Real.pi_pos; linarith)).exists |
| obtain ⟨m, hm⟩ := mem_range_of_exists_le_of_exists_ge (continuous_score μ) |
| (hneg.imp fun _ h => h.le) (hpos.imp fun _ h => h.le) |
| refine ⟨m, hm, ?_⟩ |
| intro t ht |
| exact (strictAnti_score μ).injective (ht.trans hm.symm) |
|
|
| /-- The center is a function of the unconditional probability law alone. -/ |
| def center (μ : ProbabilityMeasure ℝ) : ℝ := (existsUnique_center μ).choose |
|
|
| @[simp] theorem score_center (μ : ProbabilityMeasure ℝ) : score μ (center μ) = 0 := |
| (existsUnique_center μ).choose_spec.1 |
|
|
| lemma center_eq_of_score_eq_zero (μ : ProbabilityMeasure ℝ) {t : ℝ} |
| (ht : score μ t = 0) : center μ = t := |
| (strictAnti_score μ).injective ((score_center μ).trans ht.symm) |
|
|
| lemma score_pos_iff (μ : ProbabilityMeasure ℝ) (t : ℝ) : |
| 0 < score μ t ↔ t < center μ := by |
| rw [← score_center μ, (strictAnti_score μ).lt_iff_gt] |
|
|
| lemma score_neg_iff (μ : ProbabilityMeasure ℝ) (t : ℝ) : |
| score μ t < 0 ↔ center μ < t := by |
| rw [← score_center μ, (strictAnti_score μ).lt_iff_gt] |
|
|
| /-- Push a probability law forward by translation by `c`. -/ |
| def translate (μ : ProbabilityMeasure ℝ) (c : ℝ) : ProbabilityMeasure ℝ := |
| μ.map (fun y => y + c) |
|
|
| lemma score_translate (μ : ProbabilityMeasure ℝ) (c t : ℝ) : |
| score (translate μ c) (t + c) = score μ t := by |
| unfold score translate |
| rw [ProbabilityMeasure.toMeasure_map, integral_map_of_stronglyMeasurable |
| (measurable_id'.add_const c)] |
| · congr 1 |
| ext y |
| congr 1 |
| ring |
| · exact (Real.continuous_arctan.comp |
| (continuous_id.sub continuous_const)).stronglyMeasurable |
|
|
| /-- Translation equivariance in `measure:center`. -/ |
| theorem center_translate (μ : ProbabilityMeasure ℝ) (c : ℝ) : |
| center (translate μ c) = center μ + c := by |
| apply center_eq_of_score_eq_zero |
| rw [score_translate, score_center] |
|
|
| /-- A bounded continuous arctangent test of a law. -/ |
| def arctanTest (t : ℝ) : ℝ →ᵇ ℝ := |
| BoundedContinuousFunction.mkOfBound |
| ⟨fun y => Real.arctan (y - t), |
| Real.continuous_arctan.comp (continuous_id.sub continuous_const)⟩ Real.pi |
| (fun x y => by |
| calc |
| dist (Real.arctan (x - t)) (Real.arctan (y - t)) |
| ≤ ‖Real.arctan (x - t)‖ + ‖Real.arctan (y - t)‖ := dist_le_norm_add_norm _ _ |
| _ ≤ Real.pi / 2 + Real.pi / 2 := add_le_add (norm_arctan_le _) (norm_arctan_le _) |
| _ = Real.pi := by ring) |
|
|
| lemma continuous_score_law (t : ℝ) : Continuous (fun μ : ProbabilityMeasure ℝ => score μ t) := |
| ProbabilityMeasure.continuous_integral_boundedContinuousFunction (arctanTest t) |
|
|
| /-- Continuity under weak convergence of real probability laws in |
| `measure:center`. The filter is arbitrary, not just a filter on subsequences. -/ |
| theorem tendsto_center {ι : Type*} {l : Filter ι} {μs : ι → ProbabilityMeasure ℝ} |
| {μ : ProbabilityMeasure ℝ} (h : Tendsto μs l (𝓝 μ)) : |
| Tendsto (fun i => center (μs i)) l (𝓝 (center μ)) := by |
| refine tendsto_order.mpr ⟨?_, ?_⟩ |
| · intro a ha |
| have ht := ((continuous_score_law a).tendsto μ).comp h |
| have he := ht.eventually_const_lt ((score_pos_iff μ a).mpr ha) |
| filter_upwards [he] with i hi |
| exact (score_pos_iff (μs i) a).mp hi |
| · intro b hb |
| have ht := ((continuous_score_law b).tendsto μ).comp h |
| have he := ht.eventually_lt_const ((score_neg_iff μ b).mpr hb) |
| filter_upwards [he] with i hi |
| exact (score_neg_iff (μs i) b).mp hi |
|
|
| /-- The no-moment arctangent center is continuous for the usual weak topology |
| on probability measures. -/ |
| theorem continuous_center : Continuous center := by |
| rw [continuous_iff_continuousAt] |
| intro μ |
| exact tendsto_center tendsto_id |
|
|
| section RandomVariables |
|
|
| variable {Ω : Type*} [MeasurableSpace Ω] |
|
|
| /-- Rewriting the law-level score as an expectation on a probability space. |
| Only measurability, not integrability of the random variable, is needed. -/ |
| lemma score_map (P : ProbabilityMeasure Ω) (Y : Ω → ℝ) |
| (hY : AEMeasurable Y (P : Measure Ω)) (t : ℝ) : |
| score (P.map Y) t = ∫ ω, Real.arctan (Y ω - t) ∂(P : Measure Ω) := by |
| unfold score |
| rw [ProbabilityMeasure.toMeasure_map] |
| exact integral_map hY (Real.continuous_arctan.comp |
| (continuous_id.sub continuous_const)).aestronglyMeasurable |
|
|
| /-- The existence/uniqueness assertion of `measure:center` for an arbitrary |
| finite real random variable. The defining expectation is with respect to |
| its unconditional probability law. -/ |
| theorem existsUnique_randomVariable_center (P : ProbabilityMeasure Ω) (Y : Ω → ℝ) |
| (hY : AEMeasurable Y (P : Measure Ω)) : |
| ∃! m : ℝ, (∫ ω, Real.arctan (Y ω - m) ∂(P : Measure Ω)) = 0 := by |
| simpa only [score_map P Y hY] using existsUnique_center (P.map Y) |
|
|
| /-- Law equality implies equality of the centers; a change of probability |
| space causes no ambiguity. -/ |
| theorem center_eq_of_law_eq {Ω' : Type*} [MeasurableSpace Ω'] |
| (P : ProbabilityMeasure Ω) (Y : Ω → ℝ) (_hY : AEMeasurable Y (P : Measure Ω)) |
| (P' : ProbabilityMeasure Ω') (Y' : Ω' → ℝ) (_hY' : AEMeasurable Y' (P' : Measure Ω')) |
| (h : P.map Y = P'.map Y') : center (P.map Y) = center (P'.map Y') := |
| congrArg center h |
|
|
| end RandomVariables |
|
|
| /-! |
| The next lemmas establish only the scalar step in `measure:normalization`. |
| Here `ν` is a law of a logarithmic (truncated) diameter, and `τ` is the law |
| of the logarithm of the limiting diameter. No assertion that FK or tree |
| maps satisfy the needed distributional convergence is made. |
| -/ |
|
|
| /-- The deterministic scale `exp(m(ν) - m(τ))` from the normalization lemma. -/ |
| def diameterScale (ν τ : ProbabilityMeasure ℝ) : ℝ := |
| Real.exp (center ν - center τ) |
|
|
| lemma diameterScale_pos (ν τ : ProbabilityMeasure ℝ) : 0 < diameterScale ν τ := |
| Real.exp_pos _ |
|
|
| /-- The proposed distance rescaling is positive even when no moments exist. -/ |
| lemma inv_diameterScale_pos (ν τ : ProbabilityMeasure ℝ) : 0 < (diameterScale ν τ)⁻¹ := |
| inv_pos.mpr (diameterScale_pos ν τ) |
|
|
| /-- Along any already-established convergence of the centered log-diameter |
| laws, the deterministic scale ratio converges to the deterministic multiplier. |
| This is the scalar step in `measure:normalization`, not the subsequential |
| geometric/probabilistic input required by `thm:main`. -/ |
| theorem tendsto_diameterScale_ratio {ι : Type*} {l : Filter ι} |
| (ν : ι → ProbabilityMeasure ℝ) (τ : ProbabilityMeasure ℝ) |
| (s : ι → ℝ) (hs : ∀ i, 0 < s i) (c : ℝ) (hc : 0 < c) |
| (h : Tendsto (fun i => translate (ν i) (-Real.log (s i))) l |
| (𝓝 (translate τ (Real.log c)))) : |
| Tendsto (fun i => diameterScale (ν i) τ / s i) l (𝓝 c) := by |
| have hm := (tendsto_center h).sub_const (center τ) |
| simp only [center_translate] at hm |
| have hexp := Real.continuous_exp.continuousAt.tendsto.comp hm |
| have hterm (i : ι) : |
| Real.exp (center (ν i) + -Real.log (s i) - center τ) = diameterScale (ν i) τ / s i := by |
| rw [show center (ν i) + -Real.log (s i) - center τ = |
| (center (ν i) - center τ) - Real.log (s i) by ring] |
| rw [Real.exp_sub, Real.exp_log (hs i)] |
| rfl |
| simpa only [Function.comp_def, hterm, add_sub_cancel_left, Real.exp_log hc] using hexp |
|
|
| /-- Reciprocal scales tend to zero along such a subsequence when its |
| reference scale tends to infinity. This conclusion concerns only real |
| numbers; no GHP convergence is assumed or asserted. -/ |
| theorem tendsto_inv_diameterScale_zero {ι : Type*} {l : Filter ι} |
| (ν : ι → ProbabilityMeasure ℝ) (τ : ProbabilityMeasure ℝ) |
| (s : ι → ℝ) (hs : ∀ i, 0 < s i) (c : ℝ) (hc : 0 < c) |
| (h : Tendsto (fun i => translate (ν i) (-Real.log (s i))) l |
| (𝓝 (translate τ (Real.log c)))) (hsInf : Tendsto s l atTop) : |
| Tendsto (fun i => (diameterScale (ν i) τ)⁻¹) l (𝓝 (0 : ℝ)) := by |
| have hr := (tendsto_diameterScale_ratio ν τ s hs c hc h).inv₀ hc.ne' |
| have hz := tendsto_inv_atTop_zero.comp hsInf |
| have hp := hr.mul hz |
| have heq (i : ι) : (diameterScale (ν i) τ / s i)⁻¹ * (s i)⁻¹ = |
| (diameterScale (ν i) τ)⁻¹ := by |
| rw [inv_div, div_eq_mul_inv] |
| field_simp [(hs i).ne', (diameterScale_pos (ν i) τ).ne'] |
| simpa only [Function.comp_def, heq, mul_zero] using hp |
|
|
| end MetricMeasureLimits.DiameterCenter |
|
|
| end |
|
|
| end OAI |
|
|
