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180bc03 | 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 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 | import Mathlib
import OAI.Probability.Ballisticity.Model
namespace OAI
section
section
open MeasureTheory ProbabilityTheory Filter Function
open scoped ENNReal NNReal BigOperators Topology Classical
namespace DirectionalTransience
noncomputable def canonicalLogExcess {Ω : Type*} (A : ℝ) (p : ℕ → Ω → ℝ) (ω : Ω) : ℝ :=
(⨆ n : ℕ, ENNReal.ofReal (-Real.log (p n ω)-2*A*Real.log ((n:ℝ)+2))).toReal
lemma measurable_canonicalLogExcess {Ω : Type*} [MeasurableSpace Ω]
(A : ℝ) (p : ℕ → Ω → ℝ) (hp : ∀ n, Measurable (p n)) :
Measurable (canonicalLogExcess A p) := by
unfold canonicalLogExcess
apply Measurable.ennreal_toReal
apply Measurable.iSup
intro n
exact ((hp n).log.neg.sub_const _).ennreal_ofReal
lemma canonicalLogExcess_nonneg {Ω : Type*} (A : ℝ) (p : ℕ → Ω → ℝ) (ω : Ω) :
0 ≤ canonicalLogExcess A p ω := ENNReal.toReal_nonneg
lemma canonicalLogExcess_bounds {Ω : Type*} {A D W : ℝ} (hA : 0 ≤ A) (hD : 0 ≤ D)
(hW : 0 ≤ W) (p : ℕ → Ω → ℝ) (ω : Ω)
(hp : ∀ n, -Real.log (p n ω) ≤ A*Real.log ((n:ℝ)+2)+D*(W+1)) :
canonicalLogExcess A p ω ≤ D*(W+1) ∧
(∀ n, -Real.log (p n ω) ≤ 2*A*Real.log ((n:ℝ)+2)+canonicalLogExcess A p ω) := by
let S := ⨆ n : ℕ, ENNReal.ofReal (-Real.log (p n ω)-2*A*Real.log ((n:ℝ)+2))
have hS : S ≤ ENNReal.ofReal (D*(W+1)) := by
apply iSup_le
intro n
apply ENNReal.ofReal_le_ofReal
have hn := hp n
have hl : 0 ≤ Real.log ((n:ℝ)+2) := Real.log_nonneg (by have := Nat.cast_nonneg (α:=ℝ) n; linarith)
have hm : 0 ≤ A*Real.log ((n:ℝ)+2) := mul_nonneg hA hl
linarith
have hSt : S ≠ ⊤ := ne_top_of_le_ne_top ENNReal.ofReal_ne_top hS
constructor
· have hh := ENNReal.toReal_mono ENNReal.ofReal_ne_top hS
simpa only [canonicalLogExcess,S,ENNReal.toReal_ofReal (show 0 ≤ D*(W+1) by positivity)] using hh
· intro n
have hn : ENNReal.ofReal (-Real.log (p n ω)-2*A*Real.log ((n:ℝ)+2)) ≤ S := le_iSup (fun k : ℕ => ENNReal.ofReal (-Real.log (p k ω)-2*A*Real.log ((k:ℝ)+2))) n
have hr := (ENNReal.ofReal_le_iff_le_toReal hSt).mp hn
change -Real.log (p n ω)-2*A*Real.log ((n:ℝ)+2) ≤ canonicalLogExcess A p ω at hr
linarith
noncomputable def finiteLogExcess {Ω : Type*} (A : ℝ) (p : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) : ℝ :=
(⨆ n : Fin (N+1), ENNReal.ofReal (-Real.log (p n ω)-2*A*Real.log ((n:ℝ)+2))).toReal
lemma canonicalLogExcess_eq_finite {Ω : Type*} {A D W M : ℝ} (hA : 0 ≤ A) (hD : 0 ≤ D)
(hWM : W ≤ M) (p : ℕ → Ω → ℝ) (ω : Ω)
(hp : ∀ n, -Real.log (p n ω) ≤ A*Real.log ((n:ℝ)+2)+D*(W+1))
(N : ℕ) (hN : D*(M+1) ≤ A*Real.log ((N:ℝ)+2)) :
canonicalLogExcess A p ω = finiteLogExcess A p N ω := by
unfold canonicalLogExcess finiteLogExcess
congr 1
apply le_antisymm
· apply iSup_le
intro n
by_cases hn : n ≤ N
· exact le_iSup (fun k : Fin (N+1) => ENNReal.ofReal (-Real.log (p k ω)-2*A*Real.log ((k:ℝ)+2))) ⟨n,by omega⟩
· have hlog : Real.log ((N:ℝ)+2) ≤ Real.log ((n:ℝ)+2) :=
Real.log_le_log (by positivity) (by exact_mod_cast (show N+2 ≤ n+2 by omega))
have hAl := mul_le_mul_of_nonneg_left hlog hA
have hDW := mul_le_mul_of_nonneg_left (add_le_add_right hWM 1) hD
have hp' := hp n
have hzero : -Real.log (p n ω)-2*A*Real.log ((n:ℝ)+2) ≤ 0 := by linarith
rw [ENNReal.ofReal_eq_zero.mpr hzero]
exact zero_le
· apply iSup_le
intro n
exact le_iSup (fun k : ℕ => ENNReal.ofReal (-Real.log (p k ω)-2*A*Real.log ((k:ℝ)+2))) n.val
end DirectionalTransience
end
end
end OAI
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