| import OAI.Probability.LatticePassage.PositiveMoment |
| import OAI.Probability.LatticePassage.LocalizationMean |
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| namespace OAI |
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| namespace CompanionFPP |
| open MeasureTheory ProbabilityTheory Real |
| open scoped ENNReal |
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| lemma exponential_weighted_density (a x : ℝ) (ha : a < 1) : |
| exponentialPDF 1 x * ENNReal.ofReal (exp (a*x)) = |
| ENNReal.ofReal (1/(1-a)) * exponentialPDF (1-a) x := by |
| by_cases hx : 0 ≤ x |
| · rw [exponentialPDF_of_nonneg hx, exponentialPDF_of_nonneg hx, |
| ← ENNReal.ofReal_mul (by positivity), ← ENNReal.ofReal_mul (by positivity)] |
| congr 1 |
| simp only [one_mul] |
| rw [← exp_add] |
| have he : -x + a*x = -((1-a)*x) := by ring |
| rw [he] |
| field_simp [show 1-a ≠ 0 by linarith] |
| · rw [exponentialPDF_of_neg (lt_of_not_ge hx), |
| exponentialPDF_of_neg (lt_of_not_ge hx)] |
| simp |
|
|
| theorem exponential_lmoment (a : ℝ) (ha : a < 1) : |
| (∫⁻ x, ENNReal.ofReal (exp (a*x)) ∂expMeasure 1) = ENNReal.ofReal (1/(1-a)) := by |
| change (∫⁻ x, ENNReal.ofReal (exp (a*x)) ∂volume.withDensity (exponentialPDF 1)) = _ |
| rw [lintegral_withDensity_eq_lintegral_mul volume (by unfold exponentialPDF; fun_prop) |
| (by fun_prop)] |
| simp only [Pi.mul_apply, exponential_weighted_density a _ ha] |
| rw [lintegral_const_mul _ (by unfold exponentialPDF; fun_prop), |
| lintegral_exponentialPDF_eq_one (by linarith : 0 < 1-a), mul_one] |
|
|
| lemma integrable_exponential (a : ℝ) (ha : a < 1) : |
| Integrable (fun x => exp (a*x)) (expMeasure 1) := by |
| apply (lintegral_ofReal_ne_top_iff_integrable (by fun_prop) |
| (by filter_upwards [] with x; positivity)).mp |
| rw [exponential_lmoment a ha] |
| exact ENNReal.ofReal_ne_top |
|
|
| theorem integral_exponential (a : ℝ) (ha : a < 1) : |
| (∫ x, exp (a*x) ∂expMeasure 1) = 1/(1-a) := by |
| rw [integral_eq_lintegral_of_nonneg_ae |
| (by filter_upwards [] with x; positivity) (by fun_prop), exponential_lmoment a ha, |
| ENNReal.toReal_ofReal (by positivity)] |
|
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| lemma exponential_increment_square (a u : ℝ) : |
| (exp (a*u)-1)^2 = exp ((2*a)*u)-2*exp (a*u)+1 := by |
| rw [show (2*a)*u = a*u+a*u by ring, exp_add] |
| ring |
|
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| lemma integrable_exponential_increment_square (a : ℝ) (ha : 2*a < 1) : |
| Integrable (fun u => (exp (a*u)-1)^2) (expMeasure 1) := by |
| simp_rw [exponential_increment_square] |
| exact ((integrable_exponential (2*a) ha).sub |
| ((integrable_exponential a (by linarith)).const_mul 2)).add (integrable_const 1) |
|
|
| theorem integral_exponential_increment_square (a : ℝ) (ha : 2*a < 1) : |
| (∫ u, (exp (a*u)-1)^2 ∂expMeasure 1) = |
| 2*a^2/((1-2*a)*(1-a)) := by |
| simp_rw [exponential_increment_square] |
| rw [integral_add (f := fun u => exp ((2*a)*u)-2*exp (a*u)) (g := fun _ => (1 : ℝ)) ((integrable_exponential (2*a) ha).sub |
| ((integrable_exponential a (by linarith)).const_mul 2)) (integrable_const 1), |
| integral_sub (f := fun u => exp ((2*a)*u)) (g := fun u => 2*exp (a*u)) (integrable_exponential (2*a) ha) |
| ((integrable_exponential a (by linarith)).const_mul 2), integral_const_mul, |
| integral_exponential (2*a) ha, integral_exponential a (by linarith)] |
| simp only [integral_const, probReal_univ, smul_eq_mul, one_mul] |
| field_simp [show 1-a ≠ 0 by linarith, show 1-2*a ≠ 0 by linarith] |
| ring |
|
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| theorem exponential_half_increment_square_le (a : ℝ) (ha : |a| ≤ 1/2) : |
| (∫⁻ u, ENNReal.ofReal ((exp (a*u/2)-1)^2) ∂expMeasure 1) ≤ |
| ENNReal.ofReal (2*a^2) := by |
| have hb := abs_le.mp ha |
| have hp : 2*(a/2) < 1 := by linarith |
| have he (u : ℝ) : a*u/2 = (a/2)*u := by ring |
| simp_rw [he] |
| rw [← ofReal_integral_eq_lintegral_ofReal (integrable_exponential_increment_square (a/2) hp) |
| (by filter_upwards [] with u; positivity), integral_exponential_increment_square (a/2) hp] |
| apply ENNReal.ofReal_le_ofReal |
| have hd : 0 < (1-2*(a/2))*(1-a/2) := mul_pos (by linarith) (by linarith) |
| apply (div_le_iff₀ hd).mpr |
| have hden : (3/8 : ℝ) ≤ (1-2*(a/2))*(1-a/2) := by nlinarith |
| nlinarith [mul_nonneg (sq_nonneg a) (sub_nonneg.mpr hden)] |
|
|
| end CompanionFPP |
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| end OAI |
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