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f61e547 | 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 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 | import OAI.Probability.FreeIsing.Energy.EnergyLimit
import OAI.Probability.FreeIsing.Martingales.PredictableL2Completion
namespace OAI
noncomputable section
open Filter Finset
open scoped Topology
namespace Problem367.RowEnergyConvergence
theorem summable_limit_of_tsum_le
{ι : Type*} {a : ℕ → ι → ℝ} {b : ι → ℝ} {K : ℝ}
(ha : ∀ n i, 0 ≤ a n i) (hsum : ∀ n, Summable (a n))
(hlim : ∀ i, Tendsto (fun n => a n i) atTop (𝓝 (b i)))
(hbound : ∀ n, ∑' i, a n i ≤ K) :
Summable b ∧ ∑' i, b i ≤ K := by
classical
have hb (i : ι) : 0 ≤ b i :=
ge_of_tendsto (hlim i) (Eventually.of_forall fun n => ha n i)
have hpartial (s : Finset ι) : ∑ i ∈ s, b i ≤ K := by
apply le_of_tendsto (tendsto_finsetSum s fun i _ => hlim i)
exact Eventually.of_forall fun n =>
((hsum n).sum_le_tsum s (fun i _ => ha n i)).trans (hbound n)
have hs : Summable b := summable_of_sum_le hb hpartial
exact ⟨hs, hs.tsum_le_of_sum_le hpartial⟩
theorem uniform_complement_tails_of_two_stage_bound
{ι : Type*} {a : ℕ → ι → ℝ} {sets : ℕ → Finset ι}
{early : ℕ → ℝ} {late : ℕ → ℕ → ℝ}
(hsum : ∀ n, Summable (a n))
(hearly : Tendsto early atTop (𝓝 0))
(hlate : ∀ k, Tendsto (late k) atTop (𝓝 0))
(hbound : ∀ n k L,
(∑' i, a n i) - ∑ i ∈ sets L, a n i ≤ early k + late k L) :
∀ ε > 0, ∃ s : Finset ι,
∀ n, ∑' i : {i // i ∉ s}, a n i.1 ≤ ε := by
intro ε hε
obtain ⟨s, hs⟩ := Problem367.uniform_energy_tails_of_two_stage_bound
hearly hlate hbound ε hε
refine ⟨s, fun n => ?_⟩
have hsplit := (hsum n).sum_add_tsum_subtype_compl s
linarith [hs n]
theorem norm_sub_sq_le {H : Type*} [SeminormedAddCommGroup H] (x y : H) :
‖x - y‖ ^ 2 ≤ 2 * ‖x‖ ^ 2 + 2 * ‖y‖ ^ 2 := by
have h := norm_sub_le x y
have hx := norm_nonneg x
have hy := norm_nonneg y
have hd := norm_nonneg (x - y)
nlinarith [sq_nonneg (‖x‖ - ‖y‖)]
theorem tendsto_tsum_norm_sub_sq_of_uniform_tails
{ι H : Type*} [NormedAddCommGroup H]
{f : ℕ → ι → H} {g : ι → H} {K : ℝ}
(hsum : ∀ n, Summable (fun i => ‖f n i‖ ^ 2))
(hbound : ∀ n, ∑' i, ‖f n i‖ ^ 2 ≤ K)
(hcoordinate : ∀ i, Tendsto (fun n => f n i) atTop (𝓝 (g i)))
(htail : ∀ ε > 0, ∃ s : Finset ι,
∀ n, ∑' i : {i // i ∉ s}, ‖f n i.1‖ ^ 2 ≤ ε) :
Summable (fun i => ‖g i‖ ^ 2) ∧
(∀ n, Summable (fun i => ‖f n i - g i‖ ^ 2)) ∧
Tendsto (fun n => ∑' i, ‖f n i - g i‖ ^ 2) atTop (𝓝 0) := by
classical
have henergy (i : ι) : Tendsto (fun n => ‖f n i‖ ^ 2) atTop
(𝓝 (‖g i‖ ^ 2)) := (hcoordinate i).norm.pow 2
have hg := (summable_limit_of_tsum_le (fun n i => sq_nonneg ‖f n i‖)
hsum henergy hbound).1
have herror (n : ℕ) : Summable (fun i => ‖f n i - g i‖ ^ 2) :=
Summable.of_nonneg_of_le (fun i => sq_nonneg _) (fun i => norm_sub_sq_le _ _)
(((hsum n).mul_left 2).add (hg.mul_left 2))
refine ⟨hg, herror, tendsto_order.2 ⟨?_, ?_⟩⟩
· intro a ha
exact Eventually.of_forall fun n =>
ha.trans_le (tsum_nonneg fun i => sq_nonneg ‖f n i - g i‖)
· intro ε hε
obtain ⟨s, hs⟩ := htail (ε / 8) (by positivity)
have hgtail : ∑' i : {i // i ∉ s}, ‖g i.1‖ ^ 2 ≤ ε / 8 :=
(summable_limit_of_tsum_le
(a := fun n (i : {i // i ∉ s}) => ‖f n i.1‖ ^ 2)
(b := fun i : {i // i ∉ s} => ‖g i.1‖ ^ 2)
(fun n i => sq_nonneg ‖f n i.1‖)
(fun n => (hsum n).subtype (fun i => i ∉ s))
(fun i => henergy i.1) hs).2
have hfinite : Tendsto (fun n => ∑ i ∈ s, ‖f n i - g i‖ ^ 2)
atTop (𝓝 0) := by
have h := tendsto_finsetSum s (fun i _ =>
(((hcoordinate i).sub_const (g i)).norm).pow 2)
simpa only [sub_self, norm_zero, zero_pow (by decide : 2 ≠ 0), sum_const_zero]
using h
filter_upwards [(tendsto_order.1 hfinite).2 (ε / 2) (by positivity)] with n hn
have herrtail : (∑' i : {i // i ∉ s}, ‖f n i.1 - g i.1‖ ^ 2) ≤
2 * (∑' i : {i // i ∉ s}, ‖f n i.1‖ ^ 2) +
2 * (∑' i : {i // i ∉ s}, ‖g i.1‖ ^ 2) := by
calc
_ ≤ ∑' i : {i // i ∉ s},
(2 * ‖f n i.1‖ ^ 2 + 2 * ‖g i.1‖ ^ 2) :=
Summable.tsum_le_tsum (fun i => norm_sub_sq_le _ _)
((herror n).subtype _)
((((hsum n).subtype _).mul_left 2).add ((hg.subtype _).mul_left 2))
_ = _ := by
simpa only [Function.comp_apply, tsum_mul_left] using
((((hsum n).subtype (fun i => i ∉ s)).mul_left 2).tsum_add
((hg.subtype (fun i => i ∉ s)).mul_left 2))
rw [← (herror n).sum_add_tsum_subtype_compl s]
linarith [hs n]
open MeasureTheory
theorem tendsto_tsum_integral_sq_sub_of_uniform_tails
{Ω ι : Type*} [MeasurableSpace Ω] (μ : Measure Ω)
{f : ℕ → ι → Ω → ℝ} {g : ι → Ω → ℝ} {K : ℝ}
(hf : ∀ n i, MemLp (f n i) 2 μ) (hg : ∀ i, MemLp (g i) 2 μ)
(hsum : ∀ n, Summable (fun i => ∫ ω, (f n i ω) ^ 2 ∂μ))
(hbound : ∀ n, ∑' i, ∫ ω, (f n i ω) ^ 2 ∂μ ≤ K)
(hcoordinate : ∀ i, Tendsto
(fun n => ∫ ω, (f n i ω - g i ω) ^ 2 ∂μ) atTop (𝓝 0))
(htail : ∀ ε > 0, ∃ s : Finset ι,
∀ n, ∑' i : {i // i ∉ s}, ∫ ω, (f n i.1 ω) ^ 2 ∂μ ≤ ε) :
Summable (fun i => ∫ ω, (g i ω) ^ 2 ∂μ) ∧
(∀ n, Summable (fun i => ∫ ω, (f n i ω - g i ω) ^ 2 ∂μ)) ∧
Tendsto (fun n => ∑' i, ∫ ω, (f n i ω - g i ω) ^ 2 ∂μ)
atTop (𝓝 0) := by
let F : ℕ → ι → Lp ℝ 2 μ := fun n i => (hf n i).toLp (f n i)
let G : ι → Lp ℝ 2 μ := fun i => (hg i).toLp (g i)
have hF (n : ℕ) (i : ι) : ‖F n i‖ ^ 2 = ∫ ω, (f n i ω) ^ 2 ∂μ :=
PredictableL2.norm_toLp_sq (hf n i)
have hG (i : ι) : ‖G i‖ ^ 2 = ∫ ω, (g i ω) ^ 2 ∂μ :=
PredictableL2.norm_toLp_sq (hg i)
have hFG (n : ℕ) (i : ι) :
‖F n i - G i‖ ^ 2 = ∫ ω, (f n i ω - g i ω) ^ 2 ∂μ := by
change ‖(hf n i).toLp (f n i) - (hg i).toLp (g i)‖ ^ 2 = _
rw [← MemLp.toLp_sub]
exact PredictableL2.norm_toLp_sq ((hf n i).sub (hg i))
have hconv (i : ι) : Tendsto (fun n => F n i) atTop (𝓝 (G i)) := by
apply tendsto_iff_norm_sub_tendsto_zero.2
have h : Tendsto (fun n => ‖F n i - G i‖ ^ 2) atTop (𝓝 0) := by
simpa only [hFG] using hcoordinate i
simpa only [Real.sqrt_sq_eq_abs, abs_norm, Real.sqrt_zero] using h.sqrt
have h := tendsto_tsum_norm_sub_sq_of_uniform_tails
(f := F) (g := G) (K := K)
(fun n => by simpa only [hF] using hsum n)
(fun n => by simpa only [hF] using hbound n) hconv
(by simpa only [hF] using htail)
simpa only [hG, hFG] using h
end Problem367.RowEnergyConvergence
end
end OAI
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