| import OAI.Probability.FreeIsing.Energy.MixedMoments |
| import OAI.Probability.FreeIsing.Energy.MomentTail |
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| namespace OAI |
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| open MeasureTheory ProbabilityTheory |
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| namespace Problem367 |
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| 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 |
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| 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 |
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| 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 _)) |
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| 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 _)) |
|
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| 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) |
|
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| 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 |
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| end OAI |
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