File size: 2,594 Bytes
a89d282 | 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 | import OAI.Probability.InvariantIsing.Gaussian.GaussianFourthMoment
namespace OAI
/-! Exact mean and variance of a diagonal Gaussian quadratic form. -/
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped BigOperators
namespace InvariantIsing
def gaussianDiagonalQuadratic {N : ℕ} (a : Fin N → ℝ) (g : Fin N → ℝ) : ℝ := ∑ i, a i*(g i)^2
lemma gaussianDiagonalQuadratic_memLp {N : ℕ} (a : Fin N → ℝ) :
MemLp (gaussianDiagonalQuadratic a) 2 (Measure.pi (fun _ : Fin N => gaussianReal 0 1)) := by
exact memLp_finsetSum Finset.univ (fun i _ =>
(gaussianReal_square_memLp.comp_measurePreserving
(measurePreserving_eval (fun _ : Fin N => gaussianReal 0 1) i)).const_mul (a i))
lemma gaussianDiagonalQuadratic_integrable {N : ℕ} (a : Fin N → ℝ) :
Integrable (gaussianDiagonalQuadratic a) (Measure.pi (fun _ : Fin N => gaussianReal 0 1)) := by
apply integrable_finsetSum
intro i _
exact ((gaussianReal_square_memLp.comp_measurePreserving
(measurePreserving_eval (fun _ : Fin N => gaussianReal 0 1) i)).const_mul (a i)).integrable (by norm_num)
lemma gaussianDiagonalQuadratic_mean {N : ℕ} (a : Fin N → ℝ) :
(∫ g, gaussianDiagonalQuadratic a g ∂Measure.pi (fun _ : Fin N => gaussianReal 0 1)) = ∑ i, a i := by
unfold gaussianDiagonalQuadratic
rw [integral_finsetSum]
· apply Finset.sum_congr rfl
intro i _
rw [integral_const_mul]
have h := (measurePreserving_eval (fun _ : Fin N => gaussianReal 0 1) i).hasLaw.integral_comp
(f := fun x : ℝ => x^2) (by fun_prop)
simpa only [Function.comp_def,Function.eval,gaussianReal_second_moment,mul_one] using
congrArg (fun x : ℝ => a i*x) h
· intro i _
exact ((gaussianReal_square_memLp.comp_measurePreserving
(measurePreserving_eval (fun _ : Fin N => gaussianReal 0 1) i)).const_mul (a i)).integrable (by norm_num)
lemma gaussianDiagonalQuadratic_variance {N : ℕ} (a : Fin N → ℝ) :
variance (gaussianDiagonalQuadratic a) (Measure.pi (fun _ : Fin N => gaussianReal 0 1)) =
2*∑ i, (a i)^2 := by
have h := variance_sum_pi (μ := fun _ : Fin N => gaussianReal 0 1)
(X := fun i x => a i*x^2) (fun i => gaussianReal_square_memLp.const_mul (a i))
have he : (∑ i : Fin N, fun g : Fin N → ℝ => a i*(g i)^2) = gaussianDiagonalQuadratic a := by
funext g
simp only [Finset.sum_apply,gaussianDiagonalQuadratic]
rw [he] at h
rw [h,Finset.mul_sum]
apply Finset.sum_congr rfl
intro i _
rw [variance_const_mul,gaussianReal_square_variance]
ring
end InvariantIsing
end
end OAI
|