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f61e547 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 | import OAI.Probability.FreeIsing.Energy.MixedMoments
import OAI.Probability.FreeIsing.Energy.MomentTail
namespace OAI
open MeasureTheory ProbabilityTheory
namespace Problem367
theorem critical_cavity_moment_step
{Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ]
{b : ℕ} {theta K t : ℝ}
(htheta : 0 ≤ theta) (htheta1 : theta < 1)
(hcritical : (b : ℝ) * theta^2 = 1) (hK : 0 ≤ K) (ht : 0 < t)
(hlog : Real.log ((b : ℝ) + 1) ≤ 2 * K)
(hsmall : Real.exp (-9 * K / 2) + ((b : ℝ) + 1)^3 * Real.exp (-4 * K) ≤ 1 - theta)
(A : ℕ → ℝ) {p : ℕ} (hp : 3 ≤ p)
{X : Fin (b + 1) → Ω → ℝ} (hX : iIndepFun X μ)
(hmX : ∀ i, Measurable (X i))
(hMom : ∀ i (k : ℕ), k ≤ p → Integrable (fun ω => |X i ω|^k) μ)
(hRec : A p ≤ ∫ ω, |∑ i, X i ω|^p ∂μ)
(hOwn : ∀ k : ℕ, k ≤ p → (∫ ω, |X 0 ω|^k ∂μ) ≤
Real.exp (K * (k : ℝ)^2 / 2) * t ^ ((k : ℝ) / 4))
(hChild : ∀ (i : Fin b) k, k ≤ p → (∫ ω, |X i.succ ω|^k ∂μ) ≤ theta^k * A k)
(hLower : ∀ k, k < p → A k ≤ Real.exp (K * (k : ℝ)^2) * t ^ ((k : ℝ) / 4)) :
A p ≤ Real.exp (K * (p : ℝ)^2) * t ^ ((p : ℝ) / 4) := by
have hbound : ∀ (i : Fin (b + 1)) k, k < p → (∫ ω, |X i ω|^k ∂μ) ≤
Real.exp (K * (k : ℝ)^2) * t ^ ((k : ℝ) / 4) := by
intro i k hk
refine Fin.cases ?_ (fun j => ?_) i
· calc
_ ≤ Real.exp (K * (k : ℝ)^2 / 2) * t ^ ((k : ℝ) / 4) := hOwn k hk.le
_ ≤ _ := mul_le_mul_of_nonneg_right (Real.exp_le_exp.mpr (by nlinarith [mul_nonneg hK (sq_nonneg (k : ℝ))]))
(Real.rpow_nonneg ht.le _)
· calc
_ ≤ theta^k * A k := hChild j k hk.le
_ ≤ theta^k * (Real.exp (K * (k : ℝ)^2) * t ^ ((k : ℝ) / 4)) :=
mul_le_mul_of_nonneg_left (hLower k hk) (pow_nonneg htheta _)
_ ≤ _ := mul_le_of_le_one_left (by positivity) (pow_le_one₀ htheta htheta1.le)
have hest := independent_abs_moment_le_pure_add_remainder hX hmX
(show 0 < p by omega) hMom hK ht hbound
have hpure : (∑ i : Fin (b + 1), ∫ ω, |X i ω|^p ∂μ) ≤
Real.exp (K * (p : ℝ)^2 / 2) * t ^ ((p : ℝ) / 4) +
(b : ℝ) * theta^p * A p := by
rw [Fin.sum_univ_succ]
have hh := Finset.sum_le_sum (fun i (_ : i ∈ (Finset.univ : Finset (Fin b))) => hChild i p (le_refl _))
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul] at hh
have := add_le_add (hOwn p (le_refl _)) hh
simpa only [mul_assoc] using this
have hrec' : (1 - (b : ℝ) * theta^p) * A p ≤
(Real.exp (K * (p : ℝ)^2) * t ^ ((p : ℝ) / 4)) *
(Real.exp (-K * (p : ℝ)^2 / 2) +
((b : ℝ) + 1)^p * Real.exp (-2 * K * ((p : ℝ) - 1))) := by
have hh := hRec.trans (hest.trans (add_le_add hpure (le_refl _)))
simp only [Fintype.card_fin, Nat.cast_add, Nat.cast_one] at hh
have he1 : Real.exp (K * (p : ℝ)^2 / 2) =
Real.exp (K * (p : ℝ)^2) * Real.exp (-K * (p : ℝ)^2 / 2) := by
rw [← Real.exp_add]
congr 1
ring
have he2 : Real.exp (K * ((p : ℝ)^2 - 2 * ((p : ℝ) - 1))) =
Real.exp (K * (p : ℝ)^2) * Real.exp (-2 * K * ((p : ℝ) - 1)) := by
rw [← Real.exp_add]
congr 1
ring
rw [he1, he2] at hh
nlinarith
apply critical_moment_absorb (Nat.cast_nonneg b) htheta htheta1 hcritical hp
(by positivity) (e := Real.exp (-K * (p : ℝ)^2 / 2) +
((b : ℝ) + 1)^p * Real.exp (-2 * K * ((p : ℝ) - 1))) _ hrec'
exact (critical_moment_error_le_order_three (by positivity) hK hlog hp).trans hsmall
theorem critical_all_moments_of_independent_recursion
{Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ]
{b : ℕ} {theta K t : ℝ}
(htheta : 0 ≤ theta) (htheta1 : theta < 1)
(hcritical : (b : ℝ) * theta^2 = 1) (hK : 0 ≤ K) (ht : 0 < t)
(hlog : Real.log ((b : ℝ) + 1) ≤ 2 * K)
(hsmall : Real.exp (-9 * K / 2) + ((b : ℝ) + 1)^3 * Real.exp (-4 * K) ≤ 1 - theta)
(A : ℕ → ℝ) {X : Fin (b + 1) → Ω → ℝ}
(hX : iIndepFun X μ) (hmX : ∀ i, Measurable (X i))
(hMom : ∀ i (k : ℕ), Integrable (fun ω => |X i ω|^k) μ)
(hRec : ∀ p, A p ≤ ∫ ω, |∑ i, X i ω|^p ∂μ)
(hOwn : ∀ k : ℕ, (∫ ω, |X 0 ω|^k ∂μ) ≤
Real.exp (K * (k : ℝ)^2 / 2) * t ^ ((k : ℝ) / 4))
(hChild : ∀ (i : Fin b) k, (∫ ω, |X i.succ ω|^k ∂μ) ≤ theta^k * A k)
(hbase : ∀ p, p < 3 → A p ≤ Real.exp (K * (p : ℝ)^2) * t ^ ((p : ℝ) / 4)) :
∀ p, A p ≤ Real.exp (K * (p : ℝ)^2) * t ^ ((p : ℝ) / 4) := by
intro p
induction p using Nat.strong_induction_on with
| h p ih =>
by_cases hp : p < 3
· exact hbase p hp
· exact critical_cavity_moment_step htheta htheta1 hcritical hK ht hlog hsmall A
(by omega) hX hmX (fun i k _ => hMom i k) (hRec p)
(fun k _ => hOwn k) (fun i k _ => hChild i k) ih
theorem exists_critical_all_moments_bound
{Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ]
{b : ℕ} {theta K0 t : ℝ}
(htheta : 0 ≤ theta) (htheta1 : theta < 1)
(hcritical : (b : ℝ) * theta^2 = 1) (hK0 : 0 ≤ K0) (ht : 0 < t)
(A : ℕ → ℝ) {X : Fin (b + 1) → Ω → ℝ}
(hX : iIndepFun X μ) (hmX : ∀ i, Measurable (X i))
(hMom : ∀ i (k : ℕ), Integrable (fun ω => |X i ω|^k) μ)
(hRec : ∀ p, A p ≤ ∫ ω, |∑ i, X i ω|^p ∂μ)
(hOwn : ∀ k : ℕ, (∫ ω, |X 0 ω|^k ∂μ) ≤
Real.exp (K0 * (k : ℝ)^2) * t ^ ((k : ℝ) / 4))
(hChild : ∀ (i : Fin b) k, (∫ ω, |X i.succ ω|^k ∂μ) ≤ theta^k * A k)
(hbase : ∀ p, p < 3 → A p ≤ Real.exp (K0 * (p : ℝ)^2) * t ^ ((p : ℝ) / 4)) :
∃ K : ℝ, K0 ≤ K ∧ 0 ≤ K ∧
∀ p, A p ≤ Real.exp (K * (p : ℝ)^2) * t ^ ((p : ℝ) / 4) := by
obtain ⟨K, hKK0, hK, hlog, hsmall⟩ :=
exists_critical_moment_constant (m := (b : ℝ) + 1) (by positivity)
(show 0 < 1 - theta by linarith) (2 * K0)
have hKK0' : K0 ≤ K := by linarith
refine ⟨K, hKK0', hK, ?_⟩
apply critical_all_moments_of_independent_recursion htheta htheta1 hcritical hK ht hlog hsmall A
hX hmX hMom hRec ?_ hChild ?_
· intro k
apply (hOwn k).trans
apply mul_le_mul_of_nonneg_right _ (Real.rpow_nonneg ht.le _)
apply Real.exp_le_exp.mpr
nlinarith [mul_nonneg (sub_nonneg.mpr hKK0) (sq_nonneg (k : ℝ))]
· intro p hp
apply (hbase p hp).trans
apply mul_le_mul_of_nonneg_right _ (Real.rpow_nonneg ht.le _)
exact Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_right hKK0' (sq_nonneg _))
theorem exists_uniform_critical_all_moments_bound
{Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ]
{b : ℕ} {theta K0 : ℝ}
(htheta : 0 ≤ theta) (htheta1 : theta < 1)
(hcritical : (b : ℝ) * theta^2 = 1) (hK0 : 0 ≤ K0)
(S : Set ℝ) (ht : ∀ t ∈ S, 0 < t)
(A : ℝ → ℕ → ℝ) (X : ℝ → Fin (b + 1) → Ω → ℝ)
(hX : ∀ t ∈ S, iIndepFun (X t) μ)
(hmX : ∀ t ∈ S, ∀ i, Measurable (X t i))
(hMom : ∀ t ∈ S, ∀ i (k : ℕ), Integrable (fun ω => |X t i ω|^k) μ)
(hRec : ∀ t ∈ S, ∀ p, A t p ≤ ∫ ω, |∑ i, X t i ω|^p ∂μ)
(hOwn : ∀ t ∈ S, ∀ k : ℕ, (∫ ω, |X t 0 ω|^k ∂μ) ≤
Real.exp (K0 * (k : ℝ)^2) * t ^ ((k : ℝ) / 4))
(hChild : ∀ t ∈ S, ∀ (i : Fin b) k,
(∫ ω, |X t i.succ ω|^k ∂μ) ≤ theta^k * A t k)
(hbase : ∀ t ∈ S, ∀ p, p < 3 →
A t p ≤ Real.exp (K0 * (p : ℝ)^2) * t ^ ((p : ℝ) / 4)) :
∃ K : ℝ, K0 ≤ K ∧ 0 ≤ K ∧
∀ t ∈ S, ∀ p, A t p ≤ Real.exp (K * (p : ℝ)^2) * t ^ ((p : ℝ) / 4) := by
obtain ⟨K, hKK0, hK, hlog, hsmall⟩ :=
exists_critical_moment_constant (m := (b : ℝ) + 1) (by positivity)
(show 0 < 1 - theta by linarith) (2 * K0)
have hKK0' : K0 ≤ K := by linarith
refine ⟨K, hKK0', hK, ?_⟩
intro t htS
apply critical_all_moments_of_independent_recursion htheta htheta1 hcritical hK
(ht t htS) hlog hsmall (A t) (hX t htS) (hmX t htS) (hMom t htS) (hRec t htS)
?_ (hChild t htS) ?_
· intro k
apply (hOwn t htS k).trans
apply mul_le_mul_of_nonneg_right _ (Real.rpow_nonneg (ht t htS).le _)
apply Real.exp_le_exp.mpr
nlinarith [mul_nonneg (sub_nonneg.mpr hKK0) (sq_nonneg (k : ℝ))]
· intro p hp
apply (hbase t htS p hp).trans
apply mul_le_mul_of_nonneg_right _ (Real.rpow_nonneg (ht t htS).le _)
exact Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_right hKK0' (sq_nonneg _))
theorem exists_uniform_critical_moment_and_tail_bound
{Ω Ξ : Type*} [MeasurableSpace Ω] [MeasurableSpace Ξ]
{μ : Measure Ω} [IsProbabilityMeasure μ] {ν : Measure Ξ} [IsProbabilityMeasure ν]
{b : ℕ} {theta K0 : ℝ}
(htheta : 0 ≤ theta) (htheta1 : theta < 1)
(hcritical : (b : ℝ) * theta^2 = 1) (hK0 : 0 < K0)
(S : Set ℝ) (ht : ∀ t ∈ S, 0 < t)
(Y : ℝ → Ω → ℝ) (X : ℝ → Fin (b + 1) → Ξ → ℝ)
(hY : ∀ t ∈ S, ∀ p : ℕ, Integrable (fun ω => |Y t ω|^p) μ)
(hX : ∀ t ∈ S, iIndepFun (X t) ν)
(hmX : ∀ t ∈ S, ∀ i, Measurable (X t i))
(hMom : ∀ t ∈ S, ∀ i (k : ℕ), Integrable (fun ω => |X t i ω|^k) ν)
(hRec : ∀ t ∈ S, ∀ p : ℕ,
(∫ ω, |Y t ω|^p ∂μ) ≤ ∫ ω, |∑ i, X t i ω|^p ∂ν)
(hOwn : ∀ t ∈ S, ∀ k : ℕ, (∫ ω, |X t 0 ω|^k ∂ν) ≤
Real.exp (K0 * (k : ℝ)^2) * t ^ ((k : ℝ) / 4))
(hChild : ∀ t ∈ S, ∀ (i : Fin b) (k : ℕ),
(∫ ω, |X t i.succ ω|^k ∂ν) ≤ theta^k * ∫ ω, |Y t ω|^k ∂μ)
(hbase : ∀ t ∈ S, ∀ p : ℕ, p < 3 →
(∫ ω, |Y t ω|^p ∂μ) ≤ Real.exp (K0 * (p : ℝ)^2) * t ^ ((p : ℝ) / 4)) :
∃ K : ℝ, K0 ≤ K ∧ 0 < K ∧
(∀ t ∈ S, ∀ p : ℕ, (∫ ω, |Y t ω|^p ∂μ) ≤
Real.exp (K * (p : ℝ)^2) * t ^ ((p : ℝ) / 4)) ∧
(∀ t ∈ S, 32 * K ≤ Real.log (1 / t) →
μ {ω | t ^ (1 / 8 : ℝ) < |Y t ω|} ≤
ENNReal.ofReal (Real.exp (-(Real.log (1 / t)^2) / (512 * K)))) := by
obtain ⟨K, hKK0, _, hbound⟩ :=
exists_uniform_critical_all_moments_bound htheta htheta1 hcritical hK0.le S ht
(fun t p => ∫ ω, |Y t ω|^p ∂μ) X hX hmX hMom hRec hOwn hChild hbase
have hK : 0 < K := hK0.trans_le hKK0
refine ⟨K, hKK0, hK, hbound, ?_⟩
intro t htS hsmall
exact measure_abs_gt_rpow_le_of_moments hK (ht t htS) hsmall
(fun p _ => hY t htS p) (fun p _ => hbound t htS p)
theorem exists_small_time_for_moment_tail {K t0 : ℝ} (ht0 : 0 < t0) :
∃ tTail : ℝ, 0 < tTail ∧ tTail ≤ t0 ∧
∀ t : ℝ, 0 < t → t ≤ tTail → 32 * K ≤ Real.log (1 / t) := by
refine ⟨min t0 (Real.exp (-32 * K)), lt_min ht0 (Real.exp_pos _), min_le_left _ _, ?_⟩
intro t ht htt
have hte : t ≤ Real.exp (-32 * K) := htt.trans (min_le_right _ _)
have hl := Real.log_le_log ht hte
rw [Real.log_exp] at hl
rw [one_div, Real.log_inv]
linarith
end Problem367
end OAI
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