File size: 2,382 Bytes
67aee1d | 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 | import OAI.Analysis.FreeEntropy.RealTail
namespace OAI
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
end
end OAI
|