| import OAI.Analysis.FreeEntropy.RealTail |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory Set Filter Finset |
| namespace FiniteEntropySeparation.Model |
| variable (p : ℕ) [NeZero p] |
| theorem cumulantConstant_pos : 0 < cumulantConstant p := by |
| unfold cumulantConstant |
| exact mul_pos (by norm_num) (Real.sqrt_pos.mpr (Real.sqrt_pos.mpr (Nat.cast_pos.mpr (NeZero.pos p)))) |
| theorem fisherDeficit_tail_integrable (hp : 2 ≤ p) (a : ℝ) (ha : 0 < a) : |
| IntegrableOn (fisherDeficit p) (Ioi a) := |
| (integrableOn_Ioi_translate a).mp (fisherDeficit_shift_integrable p hp a ha) |
| theorem fisherDeficit_tail_nonneg (a : ℝ) (ha : 0 < a) : |
| 0 ≤ ∫ u in Ioi a, fisherDeficit p u := by |
| apply setIntegral_nonneg measurableSet_Ioi |
| intro u hu |
| exact fisherDeficit_nonneg p u (lt_trans ha hu) |
| theorem fisherDeficit_tail_le (hp : 2 ≤ p) (a : ℝ) (ha : 0 < a) |
| (hKa : 2*(cumulantConstant p)^2 ≤ a) : |
| (∫ u in Ioi a, fisherDeficit p u) ≤ 2*(cumulantConstant p)^6/(3*a^3) := by |
| have hi := (inverseFourth_integrable ha).const_mul (2*(cumulantConstant p)^6) |
| have hle : (∫ u in Ioi a, fisherDeficit p u) ≤ |
| ∫ u in Ioi a, 2*(cumulantConstant p)^6 * (1/u^4) := by |
| apply integral_mono_ae (fisherDeficit_tail_integrable p hp a ha) hi |
| filter_upwards [ae_restrict_mem measurableSet_Ioi] with u hu |
| simpa only [mul_one_div] using |
| (fisherDeficit_bounds p hp u (lt_trans ha hu) (hKa.trans hu.le)).2 |
| rw [integral_const_mul, inverseFourth_integral ha] at hle |
| simpa only [mul_one_div] using hle |
| theorem chiStar_tail_formula (hp : 2 ≤ p) (a : ℝ) (ha : 0 < a) : |
| chiStar (τ p) (X p a) (auxiliaryNoise p) = |
| ((gaussianEntropy p a - (1/2 : ℝ) * ∫ u in Ioi a, fisherDeficit p u : ℝ) : EReal) := by |
| rw [chiStar_covariance_formula p hp a ha, integral_Ioi_translate] |
| theorem entropy_deficit_le (hp : 2 ≤ p) (a : ℝ) (ha : 0 < a) |
| (hKa : 2*(cumulantConstant p)^2 ≤ a) : |
| (1/2 : ℝ) * (∫ u in Ioi a, fisherDeficit p u) ≤ 1/24 := by |
| have hK := cumulantConstant_pos p |
| have hb := fisherDeficit_tail_le p hp a ha hKa |
| have hc : (2*(cumulantConstant p)^2)^3 ≤ a^3 := pow_le_pow_left₀ (by positivity) hKa 3 |
| have hd : 0 < 3*a^3 := by positivity |
| have he : 2*(cumulantConstant p)^6/(3*a^3) ≤ 1/12 := by |
| apply (div_le_iff₀ hd).mpr |
| nlinarith [hc] |
| linarith |
| end FiniteEntropySeparation.Model |
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| end |
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| end OAI |
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