File size: 3,450 Bytes
b317718 | 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 | import OAI.Probability.FreeIsing.Posterior.FilteringRepresentationClosure
namespace OAI
noncomputable section
open MeasureTheory Filter
open scoped Topology
namespace Problem367.FilteringRepresentationClosure
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
theorem inner_sum_coordinate {ι : Type*} {f : ι → E} {y : E}
(hs : HasSum f y) (ho : Pairwise fun i j => inner ℝ (f i) (f j) = 0) (i : ι) :
inner ℝ y (f i) = ‖f i‖ ^ 2 := by
have h := hs.mapL (innerSL ℝ (f i))
change HasSum (fun j => inner ℝ (f i) (f j)) (inner ℝ (f i) y) at h
have hsingle : HasSum (fun j => inner ℝ (f i) (f j)) (inner ℝ (f i) (f i)) :=
hasSum_single i (fun j hji => ho hji.symm)
rw [real_inner_comm, ← real_inner_self_eq_norm_sq]
exact h.unique hsingle
theorem eq_of_orthogonal_row_limits
{ι N : Type*} {l : Filter N} [l.NeBot]
{f : N → ι → E} {y : N → E} {g : ι → E} {z m : E}
(hs : ∀ n, HasSum (f n) (y n))
(ho : ∀ n, Pairwise fun i j => inner ℝ (f n i) (f n j) = 0)
(hf : ∀ i, Tendsto (fun n => f n i) l (𝓝 (g i)))
(hy : Tendsto y l (𝓝 z)) (hg : HasSum g m)
(he : ‖z‖ ^ 2 = ‖m‖ ^ 2) : z = m := by
have hgo : Pairwise fun i j => inner ℝ (g i) (g j) = 0 := by
intro i j hij
have h := (hf i).inner (𝕜 := ℝ) (hf j)
have hz : Tendsto (fun n => inner ℝ (f n i) (f n j)) l (𝓝 (0 : ℝ)) := by
simpa only [ho _ hij] using (tendsto_const_nhds : Tendsto (fun _ : N => (0 : ℝ)) l (𝓝 0))
exact tendsto_nhds_unique h hz
have hcross (i : ι) : inner ℝ z (g i) = ‖g i‖ ^ 2 := by
have hleft := hy.inner (𝕜 := ℝ) (hf i)
have hright := (hf i).norm.pow 2
exact tendsto_nhds_unique_of_eventuallyEq hleft hright
(Eventually.of_forall fun n => inner_sum_coordinate (hs n) (ho n) i)
have hzi := hg.mapL (innerSL ℝ z)
have hmi := hg.mapL (innerSL ℝ m)
change HasSum (fun i => inner ℝ z (g i)) (inner ℝ z m) at hzi
change HasSum (fun i => inner ℝ m (g i)) (inner ℝ m m) at hmi
simp only [hcross] at hzi
simp only [inner_sum_coordinate hg hgo] at hmi
exact eq_of_cross_energy (by simpa only [real_inner_self_eq_norm_sq] using hzi.unique hmi) he
theorem ae_eq_of_orthogonal_row_meanSquare_limits
{Ω ι N : Type*} [MeasurableSpace Ω] {P : Measure Ω}
{l : Filter N} [l.NeBot]
{X : N → ι → Ω → ℝ} {Y : N → Ω → ℝ} {I : ι → Ω → ℝ} {Z M : Ω → ℝ}
(hX : ∀ n i, MemLp (X n i) 2 P) (hY : ∀ n, MemLp (Y n) 2 P)
(hI : ∀ i, MemLp (I i) 2 P) (hZ : MemLp Z 2 P) (hM : MemLp M 2 P)
(hs : ∀ n, HasSum (fun i => (hX n i).toLp (X n i)) ((hY n).toLp (Y n)))
(ho : ∀ n, Pairwise fun i j => ∫ ω, X n i ω * X n j ω ∂P = 0)
(hc : ∀ i, Tendsto (fun n => ∫ ω, (X n i ω - I i ω) ^ 2 ∂P) l (𝓝 0))
(hy : Tendsto (fun n => ∫ ω, (Y n ω - Z ω) ^ 2 ∂P) l (𝓝 0))
(hi : HasSum (fun i => (hI i).toLp (I i)) (hM.toLp M))
(he : (∫ ω, Z ω ^ 2 ∂P) = ∫ ω, M ω ^ 2 ∂P) : Z =ᵐ[P] M := by
apply (hZ.toLp_eq_toLp_iff hM).mp
apply eq_of_orthogonal_row_limits hs
(fun n => by simpa only [PredictableL2.inner_toLp] using ho n)
(fun i => tendsto_toLp_of_integral_sq_error (fun n => hX n i) (hI i) (hc i))
(tendsto_toLp_of_integral_sq_error hY hZ hy) hi
simpa only [PredictableL2.norm_toLp_sq] using he
end Problem367.FilteringRepresentationClosure
end
end OAI
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