Theory Central_Limit_Theorem
section ‹The Central Limit Theorem›
theory Central_Limit_Theorem
imports Levy
begin
theorem (in prob_space) central_limit_theorem_zero_mean:
fixes X :: "nat ⇒ 'a ⇒ real"
and μ :: "real measure"
and σ :: real
and S :: "nat ⇒ 'a ⇒ real"
assumes X_indep: "indep_vars (λi. borel) X UNIV"
and X_mean_0: "⋀n. expectation (X n) = 0"
and σ_pos: "σ > 0"
and X_square_integrable: "⋀n. integrable M (λx. (X n x)⇧2)"
and X_variance: "⋀n. variance (X n) = σ⇧2"
and X_distrib: "⋀n. distr M borel (X n) = μ"
defines "S n ≡ λx. ∑i<n. X i x"
shows "weak_conv_m (λn. distr M borel (λx. S n x / sqrt (n * σ⇧2))) std_normal_distribution"
proof -
let ?S' = "λn x. S n x / sqrt (real n * σ⇧2)"
define φ where "φ n = char (distr M borel (?S' n))" for n
define ψ where "ψ n t = char μ (t / sqrt (σ⇧2 * n))" for n t
have X_rv [simp, measurable]: "random_variable borel (X n)" for n
using X_indep unfolding indep_vars_def2 by simp
have X_integrable [simp, intro]: "integrable M (X n)" for n
by (rule square_integrable_imp_integrable[OF _ X_square_integrable]) simp_all
interpret μ: real_distribution μ
by (subst X_distrib [symmetric, of 0], rule real_distribution_distr, simp)
have μ_integrable [simp]: "integrable μ (λx. x)"
and μ_mean_integrable [simp]: "μ.expectation (λx. x) = 0"
and μ_square_integrable [simp]: "integrable μ (λx. x^2)"
and μ_variance [simp]: "μ.expectation (λx. x^2) = σ⇧2"
using assms by (simp_all add: X_distrib [symmetric, of 0] integrable_distr_eq integral_distr)
have main: "∀⇩F n in sequentially.
cmod (φ n t - (1 + (-(t^2) / 2) / n)^n) ≤
t⇧2 / (6 * σ⇧2) * (LINT x|μ. min (6 * x⇧2) (¦t / sqrt (σ⇧2 * n)¦ * ¦x¦ ^ 3))" for t
proof (rule eventually_sequentiallyI)
fix n :: nat
assume "n ≥ nat (ceiling (t^2 / 4))"
hence n: "n ≥ t^2 / 4" by (subst nat_ceiling_le_eq [symmetric])
let ?t = "t / sqrt (σ⇧2 * n)"
define ψ' where "ψ' n i = char (distr M borel (λx. X i x / sqrt (σ⇧2 * n)))" for n i
have *: "⋀n i t. ψ' n i t = ψ n t"
unfolding ψ_def ψ'_def char_def
by (subst X_distrib [symmetric]) (auto simp: integral_distr)
have "φ n t = char (distr M borel (λx. ∑i<n. X i x / sqrt (σ⇧2 * real n))) t"
by (auto simp: φ_def S_def sum_divide_distrib ac_simps)
also have "… = (∏ i < n. ψ' n i t)"
unfolding ψ'_def
apply (rule char_distr_sum)
apply (rule indep_vars_compose2[where X=X])
apply (rule indep_vars_subset)
apply (rule X_indep)
apply auto
done
also have "… = (ψ n t)^n"
by (auto simp add: * prod_constant)
finally have φ_eq: "φ n t =(ψ n t)^n" .
have "norm (ψ n t - (1 - ?t^2 * σ⇧2 / 2)) ≤ ?t⇧2 / 6 * (LINT x|μ. min (6 * x⇧2) (¦?t¦ * ¦x¦ ^ 3))"
unfolding ψ_def by (rule μ.char_approx3, auto)
also have "?t^2 * σ⇧2 = t^2 / n"
using σ_pos by (simp add: power_divide)
also have "t^2 / n / 2 = (t^2 / 2) / n"
by simp
finally have **: "norm (ψ n t - (1 + (-(t^2) / 2) / n)) ≤
?t⇧2 / 6 * (LINT x|μ. min (6 * x⇧2) (¦?t¦ * ¦x¦ ^ 3))" by simp
have "norm (φ n t - (complex_of_real (1 + (-(t^2) / 2) / n))^n) ≤
n * norm (ψ n t - (complex_of_real (1 + (-(t^2) / 2) / n)))"
using n
by (auto intro!: norm_power_diff μ.cmod_char_le_1 abs_leI
simp del: of_real_diff simp: of_real_diff[symmetric] divide_le_eq φ_eq ψ_def)
also have "… ≤ n * (?t⇧2 / 6 * (LINT x|μ. min (6 * x⇧2) (¦?t¦ * ¦x¦ ^ 3)))"
by (rule mult_left_mono [OF **], simp)
also have "… = (t⇧2 / (6 * σ⇧2) * (LINT x|μ. min (6 * x⇧2) (¦?t¦ * ¦x¦ ^ 3)))"
using σ_pos by (simp add: field_simps min_absorb2)
finally show "norm (φ n t - (1 + (-(t^2) / 2) / n)^n) ≤
(t⇧2 / (6 * σ⇧2) * (LINT x|μ. min (6 * x⇧2) (¦?t¦ * ¦x¦ ^ 3)))"
by simp
qed
show ?thesis
proof (rule levy_continuity)
fix t
let ?t = "λn. t / sqrt (σ⇧2 * n)"
have "⋀x. (λn. min (6 * x⇧2) (¦t¦ * ¦x¦ ^ 3 / ¦sqrt (σ⇧2 * real n)¦)) ⇢ 0"
using σ_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 "(λn. LINT x|μ. min (6 * x⇧2) (¦?t n¦ * ¦x¦ ^ 3)) ⇢ (LINT x|μ. 0)"
by (intro integral_dominated_convergence [where w = "λx. 6 * x^2"]) auto
then have *: "(λn. t⇧2 / (6 * σ⇧2) * (LINT x|μ. min (6 * x⇧2) (¦?t n¦ * ¦x¦ ^ 3))) ⇢ 0"
by (simp only: integral_zero tendsto_mult_right_zero)
have "(λn. complex_of_real ((1 + (-(t^2) / 2) / n)^n)) ⇢ complex_of_real (exp (-(t^2) / 2))"
by (rule isCont_tendsto_compose [OF _ tendsto_exp_limit_sequentially]) auto
then have "(λn. φ n t) ⇢ complex_of_real (exp (-(t^2) / 2))"
by (rule Lim_transform) (rule Lim_null_comparison [OF main *])
then show "(λn. char (distr M borel (?S' n)) t) ⇢ char std_normal_distribution t"
by (simp add: φ_def char_std_normal_distribution)
qed (auto intro!: real_dist_normal_dist simp: S_def)
qed
theorem (in prob_space) central_limit_theorem:
fixes X :: "nat ⇒ 'a ⇒ real"
and μ :: "real measure"
and σ :: real
and S :: "nat ⇒ 'a ⇒ real"
assumes X_indep: "indep_vars (λi. borel) X UNIV"
and X_mean: "⋀n. expectation (X n) = m"
and σ_pos: "σ > 0"
and X_square_integrable: "⋀n. integrable M (λx. (X n x)⇧2)"
and X_variance: "⋀n. variance (X n) = σ⇧2"
and X_distrib: "⋀n. distr M borel (X n) = μ"
defines "X' i x ≡ X i x - m"
shows "weak_conv_m (λn. distr M borel (λx. (∑i<n. X' i x) / sqrt (n*σ⇧2))) std_normal_distribution"
proof (intro central_limit_theorem_zero_mean)
show "indep_vars (λi. borel) X' UNIV"
unfolding X'_def[abs_def] using X_indep by (rule indep_vars_compose2) auto
have X_rv [simp, measurable]: "random_variable borel (X n)" for n
using X_indep unfolding indep_vars_def2 by simp
have X_integrable [simp, intro]: "integrable M (X n)" for n
by (rule square_integrable_imp_integrable[OF _ X_square_integrable]) simp_all
show "expectation (X' n) = 0" for n
using X_mean by (auto simp: X'_def[abs_def] prob_space)
show "σ > 0" "integrable M (λx. (X' n x)⇧2)" "variance (X' n) = σ⇧2" for n
using ‹0 < σ› X_integrable X_mean X_square_integrable X_variance unfolding X'_def
by (auto simp: prob_space power2_diff)
show "distr M borel (X' n) = distr μ borel (λx. x - m)" for n
unfolding X_distrib[of n, symmetric] using X_integrable
by (subst distr_distr) (auto simp: X'_def[abs_def] comp_def)
qed
end