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(* Theory: Central_Limit_Theorem.thy
Authors: Jeremy Avigad, Luke Serafin
*)
section \<open>The Central Limit Theorem\<close>
theory Central_Limit_Theorem
imports Levy
begin
theorem (in prob_space) central_limit_theorem:
fixes X :: "nat \<Rightarrow> 'a \<Rightarrow> real"
and \<mu> :: "real measure"
and \<sigma> :: real
and S :: "nat \<Rightarrow> 'a \<Rightarrow> real"
assumes X_indep: "indep_vars (\<lambda>i. borel) X UNIV"
and X_integrable: "\<And>n. integrable M (X n)"
and X_mean_0: "\<And>n. expectation (X n) = 0"
and \<sigma>_pos: "\<sigma> > 0"
and X_square_integrable: "\<And>n. integrable M (\<lambda>x. (X n x)\<^sup>2)"
and X_variance: "\<And>n. variance (X n) = \<sigma>\<^sup>2"
and X_distrib: "\<And>n. distr M borel (X n) = \<mu>"
defines "S n \<equiv> \<lambda>x. \<Sum>i<n. X i x"
shows "weak_conv_m (\<lambda>n. distr M borel (\<lambda>x. S n x / sqrt (n * \<sigma>\<^sup>2))) std_normal_distribution"
proof -
let ?S' = "\<lambda>n x. S n x / sqrt (real n * \<sigma>\<^sup>2)"
def \<phi> \<equiv> "\<lambda>n. char (distr M borel (?S' n))"
def \<psi> \<equiv> "\<lambda>n t. char \<mu> (t / sqrt (\<sigma>\<^sup>2 * n))"
have X_rv [simp, measurable]: "\<And>n. random_variable borel (X n)"
using X_indep unfolding indep_vars_def2 by simp
interpret \<mu>: real_distribution \<mu>
by (subst X_distrib [symmetric, of 0], rule real_distribution_distr, simp)
(* these are equivalent to the hypotheses on X, given X_distr *)
have \<mu>_integrable [simp]: "integrable \<mu> (\<lambda>x. x)"
and \<mu>_mean_integrable [simp]: "\<mu>.expectation (\<lambda>x. x) = 0"
and \<mu>_square_integrable [simp]: "integrable \<mu> (\<lambda>x. x^2)"
and \<mu>_variance [simp]: "\<mu>.expectation (\<lambda>x. x^2) = \<sigma>\<^sup>2"
using assms by (simp_all add: X_distrib [symmetric, of 0] integrable_distr_eq integral_distr)
have main: "\<forall>\<^sub>F n in sequentially.
cmod (\<phi> n t - (1 + (-(t^2) / 2) / n)^n) \<le>
t\<^sup>2 / (6 * \<sigma>\<^sup>2) * (LINT x|\<mu>. min (6 * x\<^sup>2) (\<bar>t / sqrt (\<sigma>\<^sup>2 * n)\<bar> * \<bar>x\<bar> ^ 3))" for t
proof (rule eventually_sequentiallyI)
fix n :: nat
assume "n \<ge> nat (ceiling (t^2 / 4))"
hence n: "n \<ge> t^2 / 4" by (subst nat_ceiling_le_eq [symmetric])
let ?t = "t / sqrt (\<sigma>\<^sup>2 * n)"
def \<psi>' \<equiv> "\<lambda>n i. char (distr M borel (\<lambda>x. X i x / sqrt (\<sigma>\<^sup>2 * n)))"
have *: "\<And>n i t. \<psi>' n i t = \<psi> n t"
unfolding \<psi>_def \<psi>'_def char_def
by (subst X_distrib [symmetric]) (auto simp: integral_distr)
have "\<phi> n t = char (distr M borel (\<lambda>x. \<Sum>i<n. X i x / sqrt (\<sigma>\<^sup>2 * real n))) t"
by (auto simp: \<phi>_def S_def setsum_divide_distrib ac_simps)
also have "\<dots> = (\<Prod> i < n. \<psi>' n i t)"
unfolding \<psi>'_def
apply (rule char_distr_setsum)
apply (rule indep_vars_compose2[where X=X])
apply (rule indep_vars_subset)
apply (rule X_indep)
apply auto
done
also have "\<dots> = (\<psi> n t)^n"
by (auto simp add: * setprod_constant)
finally have \<phi>_eq: "\<phi> n t =(\<psi> n t)^n" .
have "norm (\<psi> n t - (1 - ?t^2 * \<sigma>\<^sup>2 / 2)) \<le> ?t\<^sup>2 / 6 * (LINT x|\<mu>. min (6 * x\<^sup>2) (\<bar>?t\<bar> * \<bar>x\<bar> ^ 3))"
unfolding \<psi>_def by (rule \<mu>.char_approx3, auto)
also have "?t^2 * \<sigma>\<^sup>2 = t^2 / n"
using \<sigma>_pos by (simp add: power_divide)
also have "t^2 / n / 2 = (t^2 / 2) / n"
by simp
finally have **: "norm (\<psi> n t - (1 + (-(t^2) / 2) / n)) \<le>
?t\<^sup>2 / 6 * (LINT x|\<mu>. min (6 * x\<^sup>2) (\<bar>?t\<bar> * \<bar>x\<bar> ^ 3))" by simp
have "norm (\<phi> n t - (complex_of_real (1 + (-(t^2) / 2) / n))^n) \<le>
n * norm (\<psi> n t - (complex_of_real (1 + (-(t^2) / 2) / n)))"
using n
by (auto intro!: norm_power_diff \<mu>.cmod_char_le_1 abs_leI
simp del: of_real_diff simp: of_real_diff[symmetric] divide_le_eq \<phi>_eq \<psi>_def)
also have "\<dots> \<le> n * (?t\<^sup>2 / 6 * (LINT x|\<mu>. min (6 * x\<^sup>2) (\<bar>?t\<bar> * \<bar>x\<bar> ^ 3)))"
by (rule mult_left_mono [OF **], simp)
also have "\<dots> = (t\<^sup>2 / (6 * \<sigma>\<^sup>2) * (LINT x|\<mu>. min (6 * x\<^sup>2) (\<bar>?t\<bar> * \<bar>x\<bar> ^ 3)))"
using \<sigma>_pos by (simp add: field_simps min_absorb2)
finally show "norm (\<phi> n t - (1 + (-(t^2) / 2) / n)^n) \<le>
(t\<^sup>2 / (6 * \<sigma>\<^sup>2) * (LINT x|\<mu>. min (6 * x\<^sup>2) (\<bar>?t\<bar> * \<bar>x\<bar> ^ 3)))"
by simp
qed
show ?thesis
proof (rule levy_continuity)
fix t
let ?t = "\<lambda>n. t / sqrt (\<sigma>\<^sup>2 * n)"
have "\<And>x. (\<lambda>n. min (6 * x\<^sup>2) (\<bar>t\<bar> * \<bar>x\<bar> ^ 3 / \<bar>sqrt (\<sigma>\<^sup>2 * real n)\<bar>)) \<longlonglongrightarrow> 0"
using \<sigma>_pos
by (auto simp: real_sqrt_mult min_absorb2
intro!: tendsto_min[THEN tendsto_eq_rhs] sqrt_at_top[THEN filterlim_compose]
filterlim_tendsto_pos_mult_at_top filterlim_at_top_imp_at_infinity
tendsto_divide_0 filterlim_real_sequentially)
then have "(\<lambda>n. LINT x|\<mu>. min (6 * x\<^sup>2) (\<bar>?t n\<bar> * \<bar>x\<bar> ^ 3)) \<longlonglongrightarrow> (LINT x|\<mu>. 0)"
by (intro integral_dominated_convergence [where w = "\<lambda>x. 6 * x^2"]) auto
then have *: "(\<lambda>n. t\<^sup>2 / (6 * \<sigma>\<^sup>2) * (LINT x|\<mu>. min (6 * x\<^sup>2) (\<bar>?t n\<bar> * \<bar>x\<bar> ^ 3))) \<longlonglongrightarrow> 0"
by (simp only: integral_zero tendsto_mult_right_zero)
have "(\<lambda>n. complex_of_real ((1 + (-(t^2) / 2) / n)^n)) \<longlonglongrightarrow> complex_of_real (exp (-(t^2) / 2))"
by (rule isCont_tendsto_compose [OF _ tendsto_exp_limit_sequentially]) auto
then have "(\<lambda>n. \<phi> n t) \<longlonglongrightarrow> complex_of_real (exp (-(t^2) / 2))"
by (rule Lim_transform) (rule Lim_null_comparison [OF main *])
then show "(\<lambda>n. char (distr M borel (?S' n)) t) \<longlonglongrightarrow> char std_normal_distribution t"
by (simp add: \<phi>_def char_std_normal_distribution)
qed (auto intro!: real_dist_normal_dist simp: S_def)
qed
end