| import OAI.Combinatorics.ContingencyTables.Sampling.ExactMatrixTrial |
|
|
| namespace OAI |
|
|
| /-! # Limiting output law of the executed matrix rejection loop -/ |
|
|
| namespace ContingencyTables.ExactMatrixTrial |
| open MatchingFPRAS MatchingFPRAS.PhysicalStream DensePrograms |
| open ExactIntegerChoice ResidualMixture FirstSuccess |
| open scoped BigOperators Classical |
| noncomputable section |
|
|
| lemma prefix_mass {K : ℕ} (candidates : Lookup) (w : Fin K → ℕ) |
| (hw : 0<totalWeight w) (N : ℕ) (X : MatrixCode) : |
| (OnlineIntegerRetry.prefixExperiment (f := trial) (candidates,List.ofFn w) N).mean |
| (fun o => if o=some X then 1 else 0)= |
| ∑ i : Fin K,if lookup (candidates,i.val)=X then |
| (retryIntegerLaw w hw (Nat.le_pow_clog (by decide : 1<2) (totalWeight w)) N).mass (some i) |
| else 0 := by |
| rw [prefix_mean candidates w hw,Fintype.sum_option] |
| simp only [Option.map_none,reduceCtorEq,ite_false,mul_zero,zero_add,Option.map_some, |
| Option.some.injEq,mul_ite,mul_one,mul_zero] |
|
|
| / |
| including repeated matrix codes in a candidate list. -/ |
| theorem prefix_limit {K : ℕ} (candidates : Lookup) (w : Fin K → ℕ) |
| (hw : 0<totalWeight w) (X : MatrixCode) : |
| Filter.Tendsto (fun N => |
| ((OnlineIntegerRetry.prefixExperiment (f := trial) (candidates,List.ofFn w) N).mean |
| (fun o => if o=some X then 1 else 0):ℝ)) |
| Filter.atTop (nhds ((∑ i : Fin K,if lookup (candidates,i.val)=X then normalized w i else 0:ℚ):ℝ)) := by |
| simp_rw [prefix_mass candidates w hw,Rat.cast_sum] |
| apply tendsto_finsetSum |
| intro i _ |
| by_cases hi : lookup (candidates,i.val)=X |
| · simp only [hi,ite_true] |
| exact retryIntegerLaw_limit w hw _ (clog_power_lt_twice (totalWeight w) hw) i |
| · simp only [hi,ite_false,Rat.cast_zero] |
| exact tendsto_const_nhds |
|
|
| end |
| end ContingencyTables.ExactMatrixTrial |
|
|
| end OAI |
|
|
