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a2ee19a | 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 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 | import OAI.Probability.Renewal.LocalInversion
import OAI.Probability.Renewal.Envelope
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped Topology ENNReal
namespace HoneycombLocal
lemma sigma_polynomial_bound {a u σ V : ℝ} (ha : 0 < a ∧ a < 1) (hu : 0 < u)
(hσ : 0 < σ) (hV : 0 < V) {k n : ℕ} (hk : 1 ≤ k) (hn : 1 / a ≤ n)
(hku : 1 ≤ (k : ℝ) * u ^ a) (hvar : σ ^ 2 ≤ (k : ℝ) * V * u ^ (a - 2)) :
σ ≤ max V 1 * (k : ℝ) ^ n := by
have hh := sigma_rpow_bound ha hu hσ hV hku hvar
have hH : 0 < max V 1 := lt_of_lt_of_le zero_lt_one (le_max_right _ _)
have hk0 : 0 ≤ (k : ℝ) := Nat.cast_nonneg k
have hk1 : 1 ≤ (k : ℝ) := by exact_mod_cast hk
have heq (x : ℝ) (hx : 0 ≤ x) : (x ^ a) ^ (1 / a) = x := by
rw [← Real.rpow_mul hx, mul_one_div_cancel ha.1.ne', Real.rpow_one]
have hr := Real.rpow_le_rpow (Real.rpow_nonneg hσ.le a) hh (one_div_nonneg.mpr ha.1.le)
rw [heq σ hσ.le, Real.mul_rpow (Real.rpow_pos_of_pos hH a).le hk0, heq _ hH.le] at hr
refine hr.trans (mul_le_mul_of_nonneg_left ?_ hH.le)
exact (Real.rpow_le_rpow_of_exponent_le hk1 hn).trans_eq (Real.rpow_natCast _ _)
namespace MomentData
variable {p : PMF ℕ} {a : ℝ} (M : MomentData p a)
lemma third_ratio_small {δ : ℝ} (hδ : 0 < δ) :
∃ L : ℝ, 1 ≤ L ∧ ∀ u (hu : 0 < u), u ≤ M.epsilon → ∀ k : ℕ, L ≤ (k : ℝ) * u ^ a →
∀ σ : ℝ, 0 < σ → σ ^ 2 = k * natVariance (tilt p u hu.le) →
(k : ℝ) * natThird (tilt p u hu.le) / σ ^ 3 ≤ δ := by
let B := M.thirdUpper / (M.varLower * δ)
have hB : 0 < B := div_pos M.thirdUpper_pos (mul_pos M.varLower_pos hδ)
let L := max 1 (B ^ 2 / M.varLower)
refine ⟨L, le_max_left _ _, ?_⟩
intro u hu hue k hku σ hσ hvar
have hL : B ^ 2 / M.varLower ≤ (k : ℝ) * u ^ a := (le_max_right _ _).trans hku
have hid2 : u ^ (a - 2) * u ^ 2 = u ^ a := by
rw [Real.rpow_sub hu, Real.rpow_two]
field_simp
have hv := mul_le_mul_of_nonneg_left (M.var_lower u hu hue) (Nat.cast_nonneg k)
rw [← hvar] at hv
have hsu2 : M.varLower * ((k : ℝ) * u ^ a) ≤ (σ * u) ^ 2 := by
have hh := mul_le_mul_of_nonneg_right hv (sq_nonneg u)
simpa only [mul_assoc, hid2, mul_pow, mul_left_comm M.varLower (k : ℝ)] using hh
have hB2 : B ^ 2 ≤ M.varLower * ((k : ℝ) * u ^ a) := by
have hh := (div_le_iff₀ M.varLower_pos).mp hL
nlinarith
have hBsu : B ≤ σ * u := by nlinarith [mul_pos hσ hu]
have hid3 : u ^ (a - 3) * u = u ^ (a - 2) := by
calc
_ = u ^ (a - 3) * u ^ (1 : ℝ) := by rw [Real.rpow_one]
_ = u ^ (a - 3 + 1) := (Real.rpow_add hu _ _).symm
_ = _ := by congr 1; ring
have hth := mul_le_mul_of_nonneg_right (M.third_upper u hu hue) (by positivity : 0 ≤ (k : ℝ) * u)
have hth' : ((k : ℝ) * natThird (tilt p u hu.le)) * u ≤ M.thirdUpper * ((k : ℝ) * u ^ (a - 2)) := by
have hidR : M.thirdUpper * u ^ (a - 3) * ((k : ℝ) * u) =
M.thirdUpper * ((k : ℝ) * u ^ (a - 2)) := by rw [← hid3]; ring
rw [hidR] at hth
nlinarith
have hv' := mul_le_mul_of_nonneg_left hv (div_nonneg M.thirdUpper_pos.le M.varLower_pos.le)
have hv'' : M.thirdUpper * ((k : ℝ) * u ^ (a - 2)) ≤ (M.thirdUpper / M.varLower) * σ ^ 2 := by
have hidV : (M.thirdUpper / M.varLower) * ((k : ℝ) * (M.varLower * u ^ (a - 2))) =
M.thirdUpper * ((k : ℝ) * u ^ (a - 2)) := by field_simp [M.varLower_pos.ne']
rwa [hidV] at hv'
have hT : M.thirdUpper / M.varLower ≤ δ * (σ * u) := by
have hh := mul_le_mul_of_nonneg_left hBsu hδ.le
have hid : δ * B = M.thirdUpper / M.varLower := by dsimp [B]; field_simp
rwa [hid] at hh
have herr := hth'.trans hv''
have hlast := mul_le_mul_of_nonneg_right hT (sq_nonneg σ)
apply (div_le_iff₀ (by positivity : 0 < σ ^ 3)).mpr
nlinarith [mul_pos hσ hu]
lemma variance_ge_const (ha : a < 1) {u : ℝ} (hu : 0 < u) (hu1 : u ≤ 1) (hue : u ≤ M.epsilon) :
M.varLower ≤ natVariance (tilt p u hu.le) := by
have hp : 1 ≤ u ^ (a - 2) := Real.one_le_rpow_of_pos_of_le_one_of_nonpos hu hu1 (by linarith)
have hm := mul_le_mul_of_nonneg_left hp M.varLower_pos.le
simpa only [mul_one] using hm.trans (M.var_lower u hu hue)
end MomentData
lemma uniform_tilted_local (p : PMF ℕ) (hspan : SpanOne p) {a : ℝ} (ha : 0 < a ∧ a < 1)
(hpower : BoundedPowerInput p a) :
∃ ω : ℝ, 1 ≤ ω ∧ ∃ L : ℝ, 1 ≤ L ∧ ∃ ε : ℝ, 0 < ε ∧ ∃ N : ℕ,
∀ u (hu : 0 < u), u ≤ ε → ∀ k : ℕ, N ≤ k → L ≤ (k : ℝ) * u ^ a →
∀ m : ℕ, let σ := Real.sqrt ((k : ℝ) * natVariance (tilt p u hu.le))
|((m : ℝ) - k * natMean (tilt p u hu.le)) / σ| * ω ≤ 1 →
Real.exp (-1) / (8 * Real.pi * σ) ≤ (sumLaw (tilt p u hu.le) k m).toReal := by
obtain ⟨D⟩ := annulusData_exists p ha hpower
obtain ⟨M⟩ := momentData_exists p ha hpower
obtain ⟨ρ, hρ, εg, hεg, hgap⟩ := tilt_spectral_gap p hspan ha.1 hpower D.epsilon_pos
obtain ⟨ω, hω, htail⟩ := exists_integral_tail_small (D.envelope_integrable M ha.1)
(by positivity : 0 < Real.exp (-1) / 4)
have hω0 : 0 < ω := zero_lt_one.trans_le hω
let δ := Real.exp (-1) / (8 * ω ^ 4)
have hδ : 0 < δ := by dsimp [δ]; positivity
obtain ⟨L, hL, hratio⟩ := M.third_ratio_small hδ
obtain ⟨n, hn⟩ := exists_nat_ge (1 / a)
let H := max M.varUpper 1
have hpoly : Tendsto (fun k : ℕ => 2 * Real.pi * H * ((k : ℝ) ^ n * ρ ^ k)) atTop (𝓝 0) := by
simpa using (tendsto_pow_const_mul_const_pow_of_lt_one n hρ.1 hρ.2).const_mul (2 * Real.pi * H)
have hhigh := hpoly.eventually (Iio_mem_nhds (by positivity : 0 < Real.exp (-1) / 4))
have hcast := (tendsto_natCast_atTop_atTop : Tendsto (fun k : ℕ => (k : ℝ)) atTop atTop)
have hlarge : ∀ᶠ k : ℕ in atTop,
1 ≤ k ∧ ω ^ 2 ≤ 2 * (k : ℝ) ∧ (ω / Real.pi) ^ 2 ≤ (k : ℝ) * M.varLower ∧
4 * ω ^ 5 / Real.exp (-1) ≤ (k : ℝ) := by
filter_upwards [eventually_ge_atTop (1 : ℕ), hcast.eventually_ge_atTop (ω ^ 2 / 2),
hcast.eventually_ge_atTop ((ω / Real.pi) ^ 2 / M.varLower),
hcast.eventually_ge_atTop (4 * ω ^ 5 / Real.exp (-1))] with k hk h2 hv he
exact ⟨hk, by linarith, (div_le_iff₀ M.varLower_pos).mp hv, he⟩
obtain ⟨N, hN⟩ := (hlarge.and hhigh).exists_forall_of_atTop
let ε := min D.epsilon (min M.epsilon (min εg 1))
refine ⟨ω, hω, L, hL, ε, lt_min D.epsilon_pos (lt_min M.epsilon_pos (lt_min hεg zero_lt_one)), N, ?_⟩
intro u hu hue k hk hku m σ hcenter
have hdeps : u ≤ D.epsilon ∧ u ≤ M.epsilon ∧ u ≤ εg ∧ u ≤ 1 := by
simpa only [ε, le_min_iff] using hue
obtain ⟨⟨hk1, hk2, hkv, hke⟩, hkhigh⟩ := hN k hk
have hkpos : 0 < k := lt_of_lt_of_le Nat.zero_lt_one hk1
have hkpos' : (0 : ℝ) < k := by exact_mod_cast hkpos
have hv0 := M.variance_ge_const ha.2 hu hdeps.2.2.2 hdeps.2.1
have hvpos : 0 < natVariance (tilt p u hu.le) := M.varLower_pos.trans_le hv0
have hσ : 0 < σ := Real.sqrt_pos.mpr (mul_pos hkpos' hvpos)
have hvar : σ ^ 2 = k * natVariance (tilt p u hu.le) := Real.sq_sqrt (mul_pos hkpos' hvpos).le
have hwidth : ω ≤ Real.pi * σ := by
have hh := mul_le_mul_of_nonneg_left hv0 hkpos'.le
rw [← hvar] at hh
have hsq : (ω / Real.pi) ^ 2 ≤ σ ^ 2 := hkv.trans hh
have hdiv : ω / Real.pi ≤ σ := by nlinarith [sq_nonneg σ]
exact (div_le_iff₀ Real.pi_pos).mp hdiv |>.trans_eq (mul_comm _ _)
have hσup : σ ≤ H * (k : ℝ) ^ n := by
apply sigma_polynomial_bound ha hu hσ M.varUpper_pos hk1 hn (hL.trans hku)
rw [hvar]
simpa only [mul_assoc] using mul_le_mul_of_nonneg_left (M.var_upper u hu hdeps.2.1) hkpos'.le
apply local_lower_of_envelope (tilt p u hu.le) (tilt_moment_integrable p hu 3)
hkpos m hσ hω hvar hk2 hwidth hcenter (D.envelope_integrable M ha.1)
(D.envelope_nonneg M) (pow_nonneg hρ.1 k)
· intro t ht
apply D.normalized_bound M ha hu hdeps.1 hdeps.2.1 (hL.trans hku) hσ hvar hρ.1
(hgap u hu.le hdeps.2.2.1)
exact abs_le.mpr ⟨by simpa [neg_mul] using ht.1, ht.2⟩
· exact htail
· have hm := mul_le_mul_of_nonneg_right hσup (mul_nonneg (by positivity : 0 ≤ 2 * Real.pi) (pow_nonneg hρ.1 k))
nlinarith
· have hr := hratio u hu hdeps.2.1 k hku σ hσ hvar
have hfirst : (k : ℝ) * natThird (tilt p u hu.le) / σ ^ 3 * ω ^ 4 ≤ Real.exp (-1) / 8 := by
have hh := mul_le_mul_of_nonneg_right hr (pow_nonneg hω0.le 4)
have hid : δ * ω ^ 4 = Real.exp (-1) / 8 := by dsimp [δ]; field_simp
simpa only [hid] using hh
have hsecond : ω ^ 5 / (2 * (k : ℝ)) ≤ Real.exp (-1) / 8 := by
apply (div_le_iff₀ (by positivity)).mpr
have hh := (div_le_iff₀ (Real.exp_pos (-1))).mp hke
nlinarith
have hid : 2 * ω * ((k * natThird (tilt p u hu.le) / σ ^ 3) * ω ^ 3 / 2 + ω ^ 4 / (4 * k)) =
(k * natThird (tilt p u hu.le) / σ ^ 3) * ω ^ 4 + ω ^ 5 / (2 * k) := by ring
rw [hid]
linarith
end HoneycombLocal
end
end OAI
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