| import Mathlib.Analysis.Normed.Group.Tannery |
| import Mathlib.Analysis.Normed.Module.FiniteDimension |
| import Mathlib.Tactic |
| import Mathlib.Topology.Instances.Real.Lemmas |
| import Mathlib.Topology.Sequences |
|
|
| namespace OAI |
|
|
| open Filter Finset |
| open scoped Topology |
|
|
| namespace Balanced |
| open scoped _root_.Balanced |
|
|
| theorem tendsto_tsum_abs_sub_of_probability_mass |
| {ι α : Type*} {l : Filter α} {p : α → ι → ℝ} {q : ι → ℝ} |
| (hp_nonneg : ∀ n i, 0 ≤ p n i) (hq_nonneg : ∀ i, 0 ≤ q i) |
| (hp_sum : ∀ n, HasSum (p n) 1) (hq_sum : HasSum q 1) |
| (hpoint : ∀ i, Tendsto (fun n => p n i) l (𝓝 (q i))) : |
| Tendsto (fun n => ∑' i, |p n i - q i|) l (𝓝 0) := by |
| have hmin_summable (n : α) : Summable (fun i => min (p n i) (q i)) := by |
| apply hq_sum.summable.of_nonneg_of_le |
| · intro i |
| exact le_min (hp_nonneg n i) (hq_nonneg i) |
| · intro i |
| exact min_le_right _ _ |
| have hmin : Tendsto (fun n => ∑' i, min (p n i) (q i)) l (𝓝 1) := by |
| rw [← hq_sum.tsum_eq] |
| apply tendsto_tsum_of_dominated_convergence hq_sum.summable |
| · intro i |
| simpa using (hpoint i).min (tendsto_const_nhds (x := q i)) |
| · exact Filter.Eventually.of_forall fun n i => by |
| rw [Real.norm_eq_abs, abs_of_nonneg (le_min (hp_nonneg n i) (hq_nonneg i))] |
| exact min_le_right _ _ |
| have hid (n : α) : (∑' i, |p n i - q i|) = 2 - 2 * ∑' i, min (p n i) (q i) := by |
| calc |
| (∑' i, |p n i - q i|) = ∑' i, (p n i + q i - 2 * min (p n i) (q i)) := by |
| apply tsum_congr |
| intro i |
| rcases le_total (p n i) (q i) with h | h |
| · rw [min_eq_left h, abs_of_nonpos (sub_nonpos.mpr h)] |
| ring |
| · rw [min_eq_right h, abs_of_nonneg (sub_nonneg.mpr h)] |
| ring |
| _ = 2 - 2 * ∑' i, min (p n i) (q i) := by |
| rw [((hp_sum n).summable.add hq_sum.summable).tsum_sub |
| ((hmin_summable n).mul_left 2), |
| (hp_sum n).summable.tsum_add hq_sum.summable, tsum_mul_left, |
| (hp_sum n).tsum_eq, hq_sum.tsum_eq] |
| norm_num |
| simp_rw [hid] |
| have hlim := (tendsto_const_nhds (x := (2 : ℝ))).sub |
| ((tendsto_const_nhds (x := (2 : ℝ))).mul hmin) |
| simpa using hlim |
|
|
| theorem tendsto_totalVariation_of_probability_mass |
| {ι α : Type*} {l : Filter α} {p : α → ι → ℝ} {q : ι → ℝ} |
| (hp_nonneg : ∀ n i, 0 ≤ p n i) (hq_nonneg : ∀ i, 0 ≤ q i) |
| (hp_sum : ∀ n, HasSum (p n) 1) (hq_sum : HasSum q 1) |
| (hpoint : ∀ i, Tendsto (fun n => p n i) l (𝓝 (q i))) : |
| Tendsto (fun n => (1 / 2 : ℝ) * ∑' i, |p n i - q i|) l (𝓝 0) := by |
| simpa using tendsto_const_nhds.mul |
| (tendsto_tsum_abs_sub_of_probability_mass hp_nonneg hq_nonneg hp_sum hq_sum hpoint) |
|
|
| theorem profile_tail_le_of_moment |
| {ι : Type*} (w : ι → ℕ) (p : ι → ℝ) (M : ℝ) (B : ℕ) |
| (hp : ∀ i, 0 ≤ p i) (hs : Summable p) |
| (hws : Summable (fun i => (w i : ℝ) * p i)) |
| (hM : (∑' i, (w i : ℝ) * p i) ≤ M) : |
| (∑' i : {i // B < w i}, p i) ≤ M / (B + 1 : ℝ) := by |
| have hB : (0 : ℝ) < B + 1 := by positivity |
| apply (le_div_iff₀ hB).2 |
| rw [← tsum_mul_right] |
| calc |
| (∑' i : {i // B < w i}, p i * (B + 1 : ℝ)) ≤ |
| ∑' i : {i // B < w i}, (w i : ℝ) * p i := by |
| apply Summable.tsum_le_tsum |
| · intro i |
| have hi : (B + 1 : ℝ) ≤ (w i : ℝ) := by exact_mod_cast i.property |
| nlinarith [hp i] |
| · exact (hs.subtype _).mul_right _ |
| · exact hws.subtype _ |
| _ ≤ ∑' i, (w i : ℝ) * p i := by |
| exact Summable.tsum_subtype_le _ _ (fun i => mul_nonneg (Nat.cast_nonneg _) (hp i)) hws |
| _ ≤ M := hM |
|
|
| theorem hasSum_one_of_tendsto_of_tight |
| {ι α : Type*} {l : Filter α} [l.NeBot] |
| {p : α → ι → ℝ} {q : ι → ℝ} |
| (hp_nonneg : ∀ n i, 0 ≤ p n i) (hp_sum : ∀ n, HasSum (p n) 1) |
| (hpoint : ∀ i, Tendsto (fun n => p n i) l (𝓝 (q i))) |
| (htight : ∀ ε : ℝ, 0 < ε → ∃ s : Finset ι, ∀ n, 1 - ε ≤ ∑ i ∈ s, p n i) : |
| HasSum q 1 := by |
| have hq_nonneg (i : ι) : 0 ≤ q i := |
| ge_of_tendsto (hpoint i) (Filter.Eventually.of_forall fun n => hp_nonneg n i) |
| have hfinite (s : Finset ι) : ∑ i ∈ s, q i ≤ 1 := by |
| apply le_of_tendsto (tendsto_finsetSum s fun i _ => hpoint i) |
| exact Filter.Eventually.of_forall fun n => by |
| simpa only [(hp_sum n).tsum_eq] using |
| (hp_sum n).summable.sum_le_tsum s (fun i _ => hp_nonneg n i) |
| have hq_summable : Summable q := summable_of_sum_le hq_nonneg hfinite |
| have hq_le : (∑' i, q i) ≤ 1 := hq_summable.tsum_le_of_sum_le hfinite |
| have hq_ge : 1 ≤ ∑' i, q i := by |
| apply le_of_forall_pos_le_add |
| intro ε hε |
| obtain ⟨s, hs⟩ := htight ε hε |
| have hlim : 1 - ε ≤ ∑ i ∈ s, q i := |
| ge_of_tendsto (tendsto_finsetSum s fun i _ => hpoint i) |
| (Filter.Eventually.of_forall hs) |
| have hsum := hq_summable.sum_le_tsum s (fun i _ => hq_nonneg i) |
| linarith |
| simpa only [le_antisymm hq_le hq_ge] using hq_summable.hasSum |
|
|
| theorem profile_tight_of_moment |
| {ι α : Type*} (w : ι → ℕ) |
| (hfinite : ∀ B, Set.Finite {i | w i ≤ B}) |
| {p : α → ι → ℝ} {M : ℝ} |
| (hp : ∀ n i, 0 ≤ p n i) (hs : ∀ n, HasSum (p n) 1) |
| (hws : ∀ n, Summable (fun i => (w i : ℝ) * p n i)) |
| (hM : ∀ n, (∑' i, (w i : ℝ) * p n i) ≤ M) : |
| ∀ ε : ℝ, 0 < ε → ∃ s : Finset ι, ∀ n, 1 - ε ≤ ∑ i ∈ s, p n i := by |
| classical |
| intro ε hε |
| obtain ⟨B, hB⟩ := exists_nat_gt (M / ε) |
| let s := (hfinite B).toFinset |
| have hset : (↑s : Set ι)ᶜ = {i | B < w i} := by |
| ext i |
| simp [s, not_le] |
| have hratio : M / (B + 1 : ℝ) ≤ ε := by |
| apply (div_le_iff₀ (by positivity : (0 : ℝ) < B + 1)).2 |
| have haux := (div_lt_iff₀ hε).1 hB |
| nlinarith |
| refine ⟨s, fun n => ?_⟩ |
| have htail := profile_tail_le_of_moment w (p n) M B (hp n) |
| (hs n).summable (hws n) (hM n) |
| have hsum := (hs n).summable.sum_add_tsum_compl (s := s) |
| rw [hset, (hs n).tsum_eq] at hsum |
| change (∑ i ∈ s, p n i) + (∑' i : {i // B < w i}, p n i) = 1 at hsum |
| linarith |
|
|
| theorem profile_moment_le_of_tendsto |
| {ι α : Type*} {l : Filter α} [l.NeBot] (w : ι → ℕ) |
| {p : α → ι → ℝ} {q : ι → ℝ} {M : ℝ} |
| (hp : ∀ n i, 0 ≤ p n i) |
| (hws : ∀ n, Summable (fun i => (w i : ℝ) * p n i)) |
| (hM : ∀ n, (∑' i, (w i : ℝ) * p n i) ≤ M) |
| (hpoint : ∀ i, Tendsto (fun n => p n i) l (𝓝 (q i))) : |
| Summable (fun i => (w i : ℝ) * q i) ∧ |
| (∑' i, (w i : ℝ) * q i) ≤ M := by |
| have hq (i : ι) : 0 ≤ q i := |
| ge_of_tendsto (hpoint i) (Filter.Eventually.of_forall fun n => hp n i) |
| have hpartial (s : Finset ι) : ∑ i ∈ s, (w i : ℝ) * q i ≤ M := by |
| apply le_of_tendsto (tendsto_finsetSum s fun i _ => |
| (tendsto_const_nhds (x := (w i : ℝ))).mul (hpoint i)) |
| exact Filter.Eventually.of_forall fun n => |
| ((hws n).sum_le_tsum s (fun i _ => mul_nonneg (Nat.cast_nonneg _) (hp n i))).trans (hM n) |
| have hnonneg : ∀ i, 0 ≤ (w i : ℝ) * q i := |
| fun i => mul_nonneg (Nat.cast_nonneg _) (hq i) |
| exact ⟨summable_of_sum_le hnonneg hpartial, |
| Real.tsum_le_of_sum_le hnonneg hpartial⟩ |
|
|
| theorem tendsto_selected_mass_of_probability_mass |
| {ι α : Type*} {l : Filter α} {p : α → ι → ℝ} {q : ι → ℝ} |
| (S : Set ι) |
| (hp_nonneg : ∀ n i, 0 ≤ p n i) (hq_nonneg : ∀ i, 0 ≤ q i) |
| (hp_sum : ∀ n, HasSum (p n) 1) (hq_sum : HasSum q 1) |
| (hpoint : ∀ i, Tendsto (fun n => p n i) l (𝓝 (q i))) : |
| Tendsto (fun n => ∑' i : S, p n i) l (𝓝 (∑' i : S, q i)) := by |
| have hbound (n : α) : |
| ‖(∑' i : S, p n i) - ∑' i : S, q i‖ ≤ ∑' i, |p n i - q i| := by |
| have hsub := (hp_sum n).summable.sub hq_sum.summable |
| have hdiff := ((hp_sum n).summable.subtype S).tsum_sub (hq_sum.summable.subtype S) |
| simp only [Function.comp_apply] at hdiff |
| calc |
| ‖(∑' i : S, p n i) - ∑' i : S, q i‖ = |
| ‖∑' i : Subtype S, (p n i.val - q i.val)‖ := congrArg norm hdiff.symm |
| _ ≤ ∑' i : Subtype S, ‖p n i.val - q i.val‖ := |
| norm_tsum_le_tsum_norm ((hsub.subtype S).norm) |
| _ ≤ ∑' i, |p n i - q i| := by |
| convert! Summable.tsum_subtype_le (fun i => ‖p n i - q i‖) S |
| (fun i => norm_nonneg _) hsub.norm using 1 |
| rw [tendsto_iff_norm_sub_tendsto_zero] |
| exact squeeze_zero (fun n => norm_nonneg _) hbound |
| (tendsto_tsum_abs_sub_of_probability_mass hp_nonneg hq_nonneg hp_sum hq_sum hpoint) |
|
|
| end Balanced |
|
|
| end OAI |
|
|
