| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory | |
| namespace Problem367 | |
| theorem one_add_abs_pow_le_two_exp (x : ℝ) (p : ℕ) : | |
| (1 + |x|) ^ p ≤ Real.exp ((p : ℝ) * x) + Real.exp (-(p : ℝ) * x) := by | |
| calc | |
| (1 + |x|) ^ p ≤ (Real.exp |x|) ^ p := by | |
| apply pow_le_pow_left₀ (by positivity) | |
| simpa only [add_comm] using Real.add_one_le_exp |x| | |
| _ = Real.exp ((p : ℝ) * |x|) := (Real.exp_nat_mul _ _).symm | |
| _ ≤ Real.exp ((p : ℝ) * x) + Real.exp (-(p : ℝ) * x) := by | |
| rcases le_total 0 x with hx | hx | |
| · rw [abs_of_nonneg hx] | |
| exact le_add_of_nonneg_right (Real.exp_nonneg _) | |
| · rw [abs_of_nonpos hx, mul_neg, neg_mul] | |
| exact le_add_of_nonneg_left (Real.exp_nonneg _) | |
| theorem integrable_one_add_abs_pow_gaussian (p : ℕ) : | |
| Integrable (fun x : ℝ => (1 + |x|) ^ p) (gaussianReal 0 1) := by | |
| apply ((integrable_exp_mul_gaussianReal (μ := 0) (v := 1) (p : ℝ)).add | |
| (integrable_exp_mul_gaussianReal (μ := 0) (v := 1) (-(p : ℝ)))).mono' | |
| · fun_prop | |
| · filter_upwards [] with x | |
| simpa only [Pi.add_apply, Real.norm_eq_abs, abs_of_nonneg (by positivity : 0 ≤ (1 + |x|) ^ p)] | |
| using one_add_abs_pow_le_two_exp x p | |
| theorem integral_one_add_abs_pow_gaussian_le {p : ℕ} (hp : 1 ≤ p) : | |
| (∫ x : ℝ, (1 + |x|) ^ p ∂gaussianReal 0 1) ≤ Real.exp (2 * (p : ℝ) ^ 2) := by | |
| have hpos := integrable_exp_mul_gaussianReal (μ := 0) (v := 1) (p : ℝ) | |
| have hneg := integrable_exp_mul_gaussianReal (μ := 0) (v := 1) (-(p : ℝ)) | |
| have hmgf (r : ℝ) : (∫ x : ℝ, Real.exp (r * x) ∂gaussianReal 0 1) = | |
| Real.exp (r ^ 2 / 2) := by | |
| simpa [mgf] using congrFun (mgf_fun_id_gaussianReal (μ := 0) (v := 1)) r | |
| have hp' : (1 : ℝ) ≤ p := by exact_mod_cast hp | |
| have htwo : 2 ≤ Real.exp (3 * (p : ℝ) ^ 2 / 2) := by | |
| nlinarith [Real.add_one_le_exp (3 * (p : ℝ) ^ 2 / 2), sq_nonneg ((p : ℝ) - 1)] | |
| calc | |
| (∫ x : ℝ, (1 + |x|) ^ p ∂gaussianReal 0 1) ≤ | |
| ∫ x : ℝ, Real.exp ((p : ℝ) * x) + Real.exp (-(p : ℝ) * x) | |
| ∂gaussianReal 0 1 := | |
| integral_mono (integrable_one_add_abs_pow_gaussian p) (hpos.add hneg) | |
| (fun x => one_add_abs_pow_le_two_exp x p) | |
| _ = 2 * Real.exp ((p : ℝ) ^ 2 / 2) := by | |
| rw [integral_add hpos hneg, hmgf, hmgf, neg_sq] | |
| ring | |
| _ ≤ Real.exp (3 * (p : ℝ) ^ 2 / 2) * Real.exp ((p : ℝ) ^ 2 / 2) := | |
| mul_le_mul_of_nonneg_right htwo (Real.exp_nonneg _) | |
| _ = Real.exp (2 * (p : ℝ) ^ 2) := by | |
| rw [← Real.exp_add] | |
| congr 1 | |
| ring | |
| theorem abs_gaussian_observation_le {t : ℝ} (ht : 0 ≤ t) (ht1 : t ≤ 1) (x : ℝ) : | |
| |t + Real.sqrt t * x| ≤ Real.sqrt t * (1 + |x|) := by | |
| calc | |
| |t + Real.sqrt t * x| ≤ |t| + |Real.sqrt t * x| := abs_add_le _ _ | |
| _ = t + Real.sqrt t * |x| := by rw [abs_of_nonneg ht, abs_mul, | |
| abs_of_nonneg (Real.sqrt_nonneg t)] | |
| _ ≤ Real.sqrt t + Real.sqrt t * |x| := | |
| add_le_add_left (Real.le_sqrt_self_iff.mpr ht1) _ | |
| _ = Real.sqrt t * (1 + |x|) := by ring | |
| theorem integrable_gaussian_observation_abs_pow {t : ℝ} | |
| (ht : 0 ≤ t) (ht1 : t ≤ 1) (p : ℕ) : | |
| Integrable (fun x : ℝ => |t + Real.sqrt t * x| ^ p) (gaussianReal 0 1) := by | |
| apply ((integrable_one_add_abs_pow_gaussian p).const_mul (Real.sqrt t ^ p)).mono' | |
| · fun_prop | |
| · filter_upwards [] with x | |
| rw [Real.norm_eq_abs, abs_of_nonneg (pow_nonneg (abs_nonneg _) p), ← mul_pow] | |
| exact pow_le_pow_left₀ (abs_nonneg _) (abs_gaussian_observation_le ht ht1 x) p | |
| theorem integral_gaussian_observation_abs_pow_le {t : ℝ} {p : ℕ} | |
| (ht : 0 < t) (ht1 : t ≤ 1) (hp : 1 ≤ p) : | |
| (∫ x : ℝ, |t + Real.sqrt t * x| ^ p ∂gaussianReal 0 1) ≤ | |
| Real.exp (2 * (p : ℝ) ^ 2) * t ^ ((p : ℝ) / 4) := by | |
| have hscale : Real.sqrt t ^ p ≤ t ^ ((p : ℝ) / 4) := by | |
| calc | |
| Real.sqrt t ^ p = t ^ ((p : ℝ) / 2) := by | |
| rw [Real.sqrt_eq_rpow, ← Real.rpow_mul_natCast ht.le] | |
| congr 1 | |
| ring | |
| _ ≤ t ^ ((p : ℝ) / 4) := | |
| Real.rpow_le_rpow_of_exponent_ge ht ht1 (by have : 0 ≤ (p : ℝ) := Nat.cast_nonneg p; linarith) | |
| calc | |
| (∫ x : ℝ, |t + Real.sqrt t * x| ^ p ∂gaussianReal 0 1) ≤ | |
| ∫ x : ℝ, Real.sqrt t ^ p * (1 + |x|) ^ p ∂gaussianReal 0 1 := by | |
| apply integral_mono (integrable_gaussian_observation_abs_pow ht.le ht1 p) | |
| ((integrable_one_add_abs_pow_gaussian p).const_mul _) | |
| intro x | |
| change |t + Real.sqrt t * x| ^ p ≤ Real.sqrt t ^ p * (1 + |x|) ^ p | |
| rw [← mul_pow] | |
| exact pow_le_pow_left₀ (abs_nonneg _) (abs_gaussian_observation_le ht.le ht1 x) p | |
| _ = Real.sqrt t ^ p * (∫ x : ℝ, (1 + |x|) ^ p ∂gaussianReal 0 1) := | |
| integral_const_mul _ _ | |
| _ ≤ Real.sqrt t ^ p * Real.exp (2 * (p : ℝ) ^ 2) := | |
| mul_le_mul_of_nonneg_left (integral_one_add_abs_pow_gaussian_le hp) (by positivity) | |
| _ ≤ t ^ ((p : ℝ) / 4) * Real.exp (2 * (p : ℝ) ^ 2) := | |
| mul_le_mul_of_nonneg_right hscale (Real.exp_nonneg _) | |
| _ = Real.exp (2 * (p : ℝ) ^ 2) * t ^ ((p : ℝ) / 4) := mul_comm _ _ | |
| theorem integrable_gaussian_observation_abs_pow_of_hasLaw | |
| {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} {G : Ω → ℝ} | |
| (hG : HasLaw G (gaussianReal 0 1) μ) {t : ℝ} | |
| (ht : 0 ≤ t) (ht1 : t ≤ 1) (p : ℕ) : | |
| Integrable (fun ω => |t + Real.sqrt t * G ω| ^ p) μ := by | |
| have h := integrable_gaussian_observation_abs_pow ht ht1 p | |
| rw [← hG.map_eq] at h | |
| exact h.comp_aemeasurable hG.aemeasurable | |
| theorem integral_gaussian_observation_abs_pow_le_of_hasLaw | |
| {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} {G : Ω → ℝ} | |
| (hG : HasLaw G (gaussianReal 0 1) μ) {t : ℝ} {p : ℕ} | |
| (ht : 0 < t) (ht1 : t ≤ 1) (hp : 1 ≤ p) : | |
| (∫ ω, |t + Real.sqrt t * G ω| ^ p ∂μ) ≤ | |
| Real.exp (2 * (p : ℝ) ^ 2) * t ^ ((p : ℝ) / 4) := by | |
| have heq := hG.integral_comp | |
| (f := fun x : ℝ => |t + Real.sqrt t * x| ^ p) (by fun_prop) | |
| change (∫ ω, |t + Real.sqrt t * G ω| ^ p ∂μ) = _ at heq | |
| rw [heq] | |
| exact integral_gaussian_observation_abs_pow_le ht ht1 hp | |
| end Problem367 | |
| end | |
| end OAI | |