| import OAI.Probability.Renewal.UniformLocal |
| import OAI.Probability.Renewal.Counting |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology ENNReal |
| namespace HoneycombLocal |
|
|
| lemma power_moment_ratio {u : ℝ} (hu : 0 < u) (a c d : ℝ) : |
| (c * u ^ (a - 2)) / (d * u ^ (a - 1)) = c / (d * u) := by |
| calc |
| _ = (c / d) * (u ^ (a - 2) / u ^ (a - 1)) := by ring |
| _ = (c / d) * u ^ ((a - 2) - (a - 1)) := by rw [Real.rpow_sub hu (a - 2) (a - 1)] |
| _ = _ := by rw [show (a - 2) - (a - 1) = (-1 : ℝ) by ring, Real.rpow_neg_one]; ring |
| |
| namespace MomentData |
| variable {p : PMF ℕ} {a : ℝ} (M : MomentData p a) |
|
|
| lemma mean_pos {u : ℝ} (hu : 0 < u) (hue : u ≤ M.epsilon) : |
| 0 < natMean (tilt p u hu.le) := |
| (mul_pos M.muLower_pos (Real.rpow_pos_of_pos hu _)).trans_le (M.mean_lower u hu hue) |
|
|
| lemma ratio_bounds {u : ℝ} (hu : 0 < u) (hue : u ≤ M.epsilon) : |
| M.varLower / (M.muUpper * u) ≤ natVariance (tilt p u hu.le) / natMean (tilt p u hu.le) ∧ |
| natVariance (tilt p u hu.le) / natMean (tilt p u hu.le) ≤ M.varUpper / (M.muLower * u) := by |
| have hμ := M.mean_pos hu hue |
| have hv := M.var_lower u hu hue |
| have hv0 : 0 ≤ natVariance (tilt p u hu.le) := natVariance_nonneg _ |
| constructor |
| · rw [← power_moment_ratio hu a M.varLower M.muUpper] |
| exact div_le_div₀ (by positivity [M.varLower_pos]) hv hμ (M.mean_upper u hu hue) |
| · rw [← power_moment_ratio hu a M.varUpper M.muLower] |
| exact div_le_div₀ (by positivity [M.varUpper_pos]) (M.var_upper u hu hue) (by positivity [M.muLower_pos]) (M.mean_lower u hu hue) |
|
|
| lemma center_bounds {A m : ℝ} (hA : 0 < A) (hm : 0 < m) |
| (hue : A / m ≤ M.epsilon) : |
| let u := A / m |
| let μ := natMean (tilt p u (div_pos hA hm).le) |
| let x := m / μ |
| 0 < x ∧ (A ^ (1 - a) / M.muUpper) * m ^ a ≤ x ∧ |
| A / M.muUpper ≤ x * u ^ a ∧ x * u ^ a ≤ A / M.muLower := by |
| intro u μ x |
| have hu : 0 < u := div_pos hA hm |
| have hμ : 0 < μ := M.mean_pos hu hue |
| have hl := M.mean_lower u hu hue |
| have hh := M.mean_upper u hu hue |
| have hLo : m / (M.muUpper * u ^ (a - 1)) ≤ x := |
| div_le_div_of_nonneg_left hm.le hμ hh |
| have hHi : x ≤ m / (M.muLower * u ^ (a - 1)) := |
| div_le_div_of_nonneg_left hm.le (by positivity [M.muLower_pos]) hl |
| have hid (c : ℝ) : m / (c * u ^ (a - 1)) * u ^ a = A / c := by |
| have hpow : u ^ a = u ^ (a - 1) * u := by simpa using Real.rpow_add_one hu.ne' (a - 1) |
| rw [hpow] |
| dsimp [u] |
| field_simp |
| refine ⟨div_pos hm hμ, ?_, ?_, ?_⟩ |
| · have hid' : m / (M.muUpper * u ^ (a - 1)) = (A ^ (1 - a) / M.muUpper) * m ^ a := by |
| calc |
| _ = (m / (A / m) ^ (a - 1)) / M.muUpper := by dsimp [u]; ring |
| _ = _ := by rw [inverse_tilt_scale hA hm]; ring |
| rwa [hid'] at hLo |
| · have ht := mul_le_mul_of_nonneg_right hLo (Real.rpow_pos_of_pos hu a).le |
| rwa [hid] at ht |
| · have ht := mul_le_mul_of_nonneg_right hHi (Real.rpow_pos_of_pos hu a).le |
| rwa [hid] at ht |
|
|
| lemma sigma_window_bounds {A m r : ℝ} (hA : 0 < A) (hm : 0 < m) |
| (hue : A / m ≤ M.epsilon) (hr : 0 ≤ r) (hr1 : r ≤ 1) |
| {k : ℕ} (hk : k ∈ indexWindow (m / natMean (tilt p (A / m) (div_pos hA hm).le)) r) : |
| let σ := Real.sqrt ((k : ℝ) * natVariance (tilt p (A / m) (div_pos hA hm).le)) |
| Real.sqrt (M.varLower / (M.muUpper * A)) * m ≤ σ ∧ |
| σ ≤ Real.sqrt (2 * M.varUpper / (M.muLower * A)) * m := by |
| intro σ |
| let u := A / m |
| let μ := natMean (tilt p u (div_pos hA hm).le) |
| let v := natVariance (tilt p u (div_pos hA hm).le) |
| let x := m / μ |
| have hu : 0 < u := div_pos hA hm |
| have hμ : 0 < μ := M.mean_pos hu hue |
| have hx : 0 < x := div_pos hm hμ |
| obtain ⟨hklo, hkhi⟩ := mem_indexWindow hx.le hr hk |
| have hk2 : (k : ℝ) ≤ 2 * x := hkhi.trans (by nlinarith) |
| have hv : 0 ≤ v := natVariance_nonneg _ |
| have hkv : 0 ≤ (k : ℝ) * v := mul_nonneg (Nat.cast_nonneg _) hv |
| have hσ0 : 0 ≤ σ := Real.sqrt_nonneg _ |
| have hσ2 : σ ^ 2 = (k : ℝ) * v := Real.sq_sqrt hkv |
| obtain ⟨hrlo, hrhi⟩ := M.ratio_bounds hu hue |
| have hlo : M.varLower / (M.muUpper * A) * m ^ 2 ≤ σ ^ 2 := by |
| calc |
| _ = m * (M.varLower / (M.muUpper * u)) := by dsimp [u]; field_simp |
| _ ≤ m * (v / μ) := mul_le_mul_of_nonneg_left hrlo hm.le |
| _ = x * v := by dsimp [x]; ring |
| _ ≤ (k : ℝ) * v := mul_le_mul_of_nonneg_right hklo hv |
| _ = σ ^ 2 := hσ2.symm |
| have hhi : σ ^ 2 ≤ (2 * M.varUpper / (M.muLower * A)) * m ^ 2 := by |
| calc |
| _ = (k : ℝ) * v := hσ2 |
| _ ≤ 2 * x * v := mul_le_mul_of_nonneg_right hk2 hv |
| _ = 2 * m * (v / μ) := by dsimp [x]; ring |
| _ ≤ 2 * m * (M.varUpper / (M.muLower * u)) := mul_le_mul_of_nonneg_left hrhi (by positivity) |
| _ = _ := by dsimp [u]; field_simp |
| have hblo : 0 ≤ M.varLower / (M.muUpper * A) := by positivity [M.varLower_pos, M.muUpper_pos] |
| have hbhi : 0 ≤ 2 * M.varUpper / (M.muLower * A) := by positivity [M.varUpper_pos, M.muLower_pos] |
| have hsql := Real.sq_sqrt hblo |
| have hsqh := Real.sq_sqrt hbhi |
| constructor |
| · nlinarith [Real.sqrt_nonneg (M.varLower / (M.muUpper * A)), sq_nonneg σ] |
| · nlinarith [Real.sqrt_nonneg (2 * M.varUpper / (M.muLower * A)), |
| mul_nonneg (Real.sqrt_nonneg (2 * M.varUpper / (M.muLower * A))) hm.le] |
|
|
| lemma centered_window {A m r ω : ℝ} (hA : 0 < A) (hm : 0 < m) |
| (hue : A / m ≤ M.epsilon) (hr : 0 ≤ r) (hr1 : r ≤ 1) (hω : 0 ≤ ω) |
| (hrω : r * ω ≤ Real.sqrt (M.varLower / (M.muUpper * A))) |
| {k : ℕ} (hk : k ∈ indexWindow (m / natMean (tilt p (A / m) (div_pos hA hm).le)) r) : |
| let μ := natMean (tilt p (A / m) (div_pos hA hm).le) |
| let σ := Real.sqrt ((k : ℝ) * natVariance (tilt p (A / m) (div_pos hA hm).le)) |
| |(m - k * μ) / σ| * ω ≤ 1 := by |
| intro μ σ |
| have hμ : 0 < μ := M.mean_pos (div_pos hA hm) hue |
| have hx : 0 < m / μ := div_pos hm hμ |
| obtain ⟨hklo, hkhi⟩ := mem_indexWindow hx.le hr hk |
| have hlo := (M.sigma_window_bounds hA hm hue hr hr1 hk).1 |
| have hσ : 0 < σ := (mul_pos (Real.sqrt_pos.mpr (by positivity [M.varLower_pos, M.muUpper_pos])) hm).trans_le hlo |
| have hkm : m ≤ (k : ℝ) * μ := (div_le_iff₀ hμ).mp hklo |
| have hkmax : (k : ℝ) * μ ≤ (1 + r) * m := by |
| have hh := mul_le_mul_of_nonneg_right hkhi hμ.le |
| field_simp at hh |
| exact hh |
| rw [abs_div, abs_of_pos hσ, abs_of_nonpos (by linarith : m - (k : ℝ) * μ ≤ 0)] |
| have hmul := mul_le_mul_of_nonneg_right (show (k : ℝ) * μ - m ≤ r * m by linarith) hω |
| have hmul2 := mul_le_mul_of_nonneg_right hrω hm.le |
| have hb : (-(m - (k : ℝ) * μ)) * ω ≤ σ := by nlinarith |
| calc |
| _ = (-(m - (k : ℝ) * μ) * ω) / σ := by ring |
| _ ≤ 1 := (div_le_one hσ).mpr hb |
|
|
| end MomentData |
| end HoneycombLocal |
|
|
| end |
|
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| end OAI |
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