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f0f05de | 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 | import OAI.Combinatorics.BooleanFunctions.MomentTransfer
import OAI.Combinatorics.BooleanFunctions.FiniteCLT
namespace OAI
open scoped BigOperators Topology NNReal ENNReal
open MeasureTheory ProbabilityTheory Filter Real
noncomputable section
namespace SquareRootDegree
namespace ReadoutCLT
open MeasureTheory ProbabilityTheory Filter UniformFinite FiniteCLT
open scoped Topology
attribute [local instance] Classical.propDecidable
def test (x : ℝ) : ℝ := (1/2)*SmoothGaussian.tent 1 (1/2) x
def constant : ℝ := (∫ x, test x ∂gaussianReal 0 1)/2
theorem constant_pos : 0 < constant := by
have h := SmoothGaussian.tent_mass_pos (a := (1:ℝ)) (h := (1/2:ℝ)) (by norm_num)
unfold constant test
rw [integral_const_mul]
positivity
theorem test_le_abs (x : ℝ) : test x ≤ |x| := by
by_cases h : |x-1| < 1/2
· have hx := (abs_lt.mp h).1
have ht := SmoothGaussian.tent_le_one (by norm_num : (0:ℝ) < 1/2) 1 x
unfold test
have ha := le_abs_self x
linarith
· rw [test,SmoothGaussian.tent_eq_zero (by norm_num : (0:ℝ) < 1/2) (le_of_not_gt h),mul_zero]
exact abs_nonneg x
variable {α : Type*} [Fintype α] [Nonempty α] [MeasurableSpace α] [MeasurableSingletonClass α]
theorem exists_abs_gain (T : α → ℝ) (h0 : average T = 0)
(h1 : average (fun a => (T a)^2) = 1) :
∃ n : ℕ, 0 < n ∧ constant < average (fun x => |normalizedSum T n x|) := by
have hc := MomentTransfer.compact_test_tendsto
(fun n => uniformMeasure (Fin n → α)) (normalizedSum T) (uniform_clt T h0 h1)
test (by unfold test; fun_prop)
((SmoothGaussian.tent_compactSupport (a := (1:ℝ)) (by norm_num : (0:ℝ) < 1/2)).mul_left (f := fun _ : ℝ => (1/2:ℝ)))
have hl : constant < ∫ x, test x ∂gaussianReal 0 1 := by
have := constant_pos
unfold constant at *
linarith
obtain ⟨n,hn,hg⟩ := ((eventually_gt_atTop 0).and (hc.eventually (lt_mem_nhds hl))).exists
refine ⟨n,hn,hg.trans_le ?_⟩
rw [integral_uniform]
exact average_mono (fun x => test_le_abs (normalizedSum T n x))
end ReadoutCLT
end SquareRootDegree
end
end OAI
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