# Theory Weak_Convergence

```(*  Title:     HOL/Probability/Weak_Convergence.thy
Authors:   Jeremy Avigad (CMU), Johannes Hölzl (TUM)
*)

section ‹Weak Convergence of Functions and Distributions›

text ‹Properties of weak convergence of functions and measures, including the portmanteau theorem.›

theory Weak_Convergence
imports Distribution_Functions
begin

section ‹Weak Convergence of Functions›

definition
weak_conv :: "(nat ⇒ (real ⇒ real)) ⇒ (real ⇒ real) ⇒ bool"
where
"weak_conv F_seq F ≡ ∀x. isCont F x ⟶ (λn. F_seq n x) ⇢ F x"

section ‹Weak Convergence of Distributions›

definition
weak_conv_m :: "(nat ⇒ real measure) ⇒ real measure ⇒ bool"
where
"weak_conv_m M_seq M ≡ weak_conv (λn. cdf (M_seq n)) (cdf M)"

section ‹Skorohod's theorem›

locale right_continuous_mono =
fixes f :: "real ⇒ real" and a b :: real
assumes cont: "⋀x. continuous (at_right x) f"
assumes mono: "mono f"
assumes bot: "(f ⤏ a) at_bot"
assumes top: "(f ⤏ b) at_top"
begin

abbreviation I :: "real ⇒ real" where
"I ω ≡ Inf {x. ω ≤ f x}"

lemma pseudoinverse: assumes "a < ω" "ω < b" shows "ω ≤ f x ⟷ I ω ≤ x"
proof
let ?F = "{x. ω ≤ f x}"
obtain y where "f y < ω"
by (metis eventually_happens' trivial_limit_at_bot_linorder order_tendstoD(2) bot ‹a < ω›)
with mono have bdd: "bdd_below ?F"
by (auto intro!: bdd_belowI[of _ y] elim: mono_invE[OF _ less_le_trans])

have ne: "?F ≠ {}"
using order_tendstoD(1)[OF top ‹ω < b›]
by (auto dest!: eventually_happens'[OF trivial_limit_at_top_linorder] intro: less_imp_le)

show "ω ≤ f x ⟹ I ω ≤ x"
by (auto intro!: cInf_lower bdd)

{ assume *: "I ω ≤ x"
have "ω ≤ (INF s∈{x. ω ≤ f x}. f s)"
by (rule cINF_greatest[OF ne]) auto
also have "… = f (I ω)"
using continuous_at_Inf_mono[OF mono cont ne bdd] ..
also have "… ≤ f x"
using * by (rule monoD[OF ‹mono f›])
finally show "ω ≤ f x" . }
qed

lemma pseudoinverse': "∀ω∈{a<..<b}. ∀x. ω ≤ f x ⟷ I ω ≤ x"
by (intro ballI allI impI pseudoinverse) auto

lemma mono_I: "mono_on {a <..< b} I"
unfolding mono_on_def by (metis order.trans order.refl pseudoinverse')

end

locale cdf_distribution = real_distribution
begin

abbreviation "C ≡ cdf M"

sublocale right_continuous_mono C 0 1
by standard
(auto intro: cdf_nondecreasing cdf_is_right_cont cdf_lim_at_top_prob cdf_lim_at_bot monoI)

lemma measurable_C[measurable]: "C ∈ borel_measurable borel"
by (intro borel_measurable_mono mono)

lemma measurable_CI[measurable]: "I ∈ borel_measurable (restrict_space borel {0<..<1})"
by (intro borel_measurable_mono_on_fnc mono_I)

lemma emeasure_distr_I: "emeasure (distr (restrict_space lborel {0<..<1::real}) borel I) UNIV = 1"
by (simp add: emeasure_distr space_restrict_space emeasure_restrict_space )

lemma distr_I_eq_M: "distr (restrict_space lborel {0<..<1::real}) borel I = M" (is "?I = _")
proof (intro cdf_unique ext)
let ?Ω = "restrict_space lborel {0<..<1}::real measure"
interpret Ω: prob_space ?Ω
by (auto simp add: emeasure_restrict_space space_restrict_space intro!: prob_spaceI)
show "real_distribution ?I"
by auto

fix x
have "cdf ?I x = measure lborel {ω∈{0<..<1}. ω ≤ C x}"
by (subst cdf_def)
(auto simp: pseudoinverse[symmetric] measure_distr space_restrict_space measure_restrict_space
intro!: arg_cong2[where f="measure"])
also have "… = measure lborel {0 <..< C x}"
using cdf_bounded_prob[of x] AE_lborel_singleton[of "C x"]
by (auto intro!: arg_cong[where f=enn2real] emeasure_eq_AE simp: measure_def)
also have "… = C x"
by (simp add: cdf_nonneg)
finally show "cdf (distr ?Ω borel I) x = C x" .
qed standard

end

context
fixes μ :: "nat ⇒ real measure"
and M :: "real measure"
assumes μ: "⋀n. real_distribution (μ n)"
assumes M: "real_distribution M"
assumes μ_to_M: "weak_conv_m μ M"
begin

(* state using obtains? *)
theorem Skorohod:
"∃ (Ω :: real measure) (Y_seq :: nat ⇒ real ⇒ real) (Y :: real ⇒ real).
prob_space Ω ∧
(∀n. Y_seq n ∈ measurable Ω borel) ∧
(∀n. distr Ω borel (Y_seq n) = μ n) ∧
Y ∈ measurable Ω lborel ∧
distr Ω borel Y = M ∧
(∀x ∈ space Ω. (λn. Y_seq n x) ⇢ Y x)"
proof -
interpret μ: cdf_distribution "μ n" for n
unfolding cdf_distribution_def by (rule μ)
interpret M: cdf_distribution M
unfolding cdf_distribution_def by (rule M)

have conv: "measure M {x} = 0 ⟹ (λn. μ.C n x) ⇢ M.C x" for x
using μ_to_M M.isCont_cdf by (auto simp: weak_conv_m_def weak_conv_def)

let ?Ω = "restrict_space lborel {0<..<1} :: real measure"
have "prob_space ?Ω"
by (auto simp: space_restrict_space emeasure_restrict_space intro!: prob_spaceI)
interpret Ω: prob_space ?Ω
by fact

have Y_distr: "distr ?Ω borel M.I = M"
by (rule M.distr_I_eq_M)

have Y_cts_cnv: "(λn. μ.I n ω) ⇢ M.I ω"
if ω: "ω ∈ {0<..<1}" "isCont M.I ω" for ω :: real
proof (intro limsup_le_liminf_real)
show "liminf (λn. μ.I n ω) ≥ M.I ω"
unfolding le_Liminf_iff
proof safe
fix B :: ereal assume B: "B < M.I ω"
then show "∀⇩F n in sequentially. B < μ.I n ω"
proof (cases B)
case (real r)
with B have r: "r < M.I ω"
by simp
then obtain x where x: "r < x" "x < M.I ω" "measure M {x} = 0"
using open_minus_countable[OF M.countable_support, of "{r<..<M.I ω}"] by auto
then have Fx_less: "M.C x < ω"
using M.pseudoinverse' ω not_less by blast

have "∀⇩F n in sequentially. μ.C n x < ω"
using order_tendstoD(2)[OF conv[OF x(3)] Fx_less] .
then have "∀⇩F n in sequentially. x < μ.I n ω"
by eventually_elim (insert ω μ.pseudoinverse[symmetric], simp add: not_le[symmetric])
then show ?thesis
by eventually_elim (insert x(1), simp add: real)
qed auto
qed

have *: "limsup (λn. μ.I n ω) ≤ M.I ω'"
if ω': "0 < ω'" "ω' < 1" "ω < ω'" for ω' :: real
proof (rule dense_ge_bounded)
fix B' assume "ereal (M.I ω') < B'" "B' < ereal (M.I ω' + 1)"
then obtain B where "M.I ω' < B" and [simp]: "B' = ereal B"
by (cases B') auto
then obtain y where y: "M.I ω' < y" "y < B" "measure M {y} = 0"
using open_minus_countable[OF M.countable_support, of "{M.I ω'<..<B}"] by auto
then have "ω' ≤ M.C (M.I ω')"
using M.pseudoinverse' ω' by (metis greaterThanLessThan_iff order_refl)
also have "... ≤ M.C y"
using M.mono y unfolding mono_def by auto
finally have Fy_gt: "ω < M.C y"
using ω'(3) by simp

have "∀⇩F n in sequentially. ω ≤ μ.C n y"
using order_tendstoD(1)[OF conv[OF y(3)] Fy_gt] by eventually_elim (rule less_imp_le)
then have 2: "∀⇩F n in sequentially. μ.I n ω ≤ ereal y"
by simp (subst μ.pseudoinverse'[rule_format, OF ω(1), symmetric])
then show "limsup (λn. μ.I n ω) ≤ B'"
using ‹y < B›
by (intro Limsup_bounded[rotated]) (auto intro: le_less_trans elim: eventually_mono)
qed simp

have **: "(M.I ⤏ ereal (M.I ω)) (at_right ω)"
using ω(2) by (auto intro: tendsto_within_subset simp: continuous_at)
show "limsup (λn. μ.I n ω) ≤ M.I ω"
using ω
by (intro tendsto_lowerbound[OF **])
(auto intro!: exI[of _ 1] * simp: eventually_at_right[of _ 1])
qed

let ?D = "{ω∈{0<..<1}. ¬ isCont M.I ω}"
have D_countable: "countable ?D"
using mono_on_ctble_discont[OF M.mono_I] by (simp add: at_within_open[of _ "{0 <..< 1}"] cong: conj_cong)
hence D: "emeasure ?Ω ?D = 0"
using emeasure_lborel_countable[OF D_countable]
by (subst emeasure_restrict_space) auto

define Y' where "Y' ω = (if ω ∈ ?D then 0 else M.I ω)" for ω
have Y'_AE: "AE ω in ?Ω. Y' ω = M.I ω"
by (rule AE_I [OF _ D]) (auto simp: space_restrict_space sets_restrict_space_iff Y'_def)

define Y_seq' where "Y_seq' n ω = (if ω ∈ ?D then 0 else μ.I n ω)" for n ω
have Y_seq'_AE: "⋀n. AE ω in ?Ω. Y_seq' n ω = μ.I n ω"
by (rule AE_I [OF _ D]) (auto simp: space_restrict_space sets_restrict_space_iff Y_seq'_def)

have Y'_cnv: "∀ω∈{0<..<1}. (λn. Y_seq' n ω) ⇢ Y' ω"
by (auto simp: Y'_def Y_seq'_def Y_cts_cnv)

have [simp]: "Y_seq' n ∈ borel_measurable ?Ω" for n
by (rule measurable_discrete_difference[of "μ.I n" _ _ ?D])
(insert μ.measurable_CI[of n] D_countable, auto simp: sets_restrict_space Y_seq'_def)
moreover have "distr ?Ω borel (Y_seq' n) = μ n" for n
using μ.distr_I_eq_M [of n] Y_seq'_AE [of n]
by (subst distr_cong_AE[where f = "Y_seq' n" and g = "μ.I n"], auto)
moreover have [simp]: "Y' ∈ borel_measurable ?Ω"
by (rule measurable_discrete_difference[of M.I _ _ ?D])
(insert M.measurable_CI D_countable, auto simp: sets_restrict_space Y'_def)
moreover have "distr ?Ω borel Y' = M"
using M.distr_I_eq_M Y'_AE
by (subst distr_cong_AE[where f = Y' and g = M.I], auto)
ultimately have "prob_space ?Ω ∧ (∀n. Y_seq' n ∈ borel_measurable ?Ω) ∧
(∀n. distr ?Ω borel (Y_seq' n) = μ n) ∧ Y' ∈ measurable ?Ω lborel ∧ distr ?Ω borel Y' = M ∧
(∀x∈space ?Ω. (λn. Y_seq' n x) ⇢ Y' x)"
using Y'_cnv ‹prob_space ?Ω› by (auto simp: space_restrict_space)
thus ?thesis by metis
qed

text ‹
The Portmanteau theorem, that is, the equivalence of various definitions of weak convergence.
›

theorem weak_conv_imp_bdd_ae_continuous_conv:
fixes
f :: "real ⇒ 'a::{banach, second_countable_topology}"
assumes
discont_null: "M ({x. ¬ isCont f x}) = 0" and
f_bdd: "⋀x. norm (f x) ≤ B" and
[measurable]: "f ∈ borel_measurable borel"
shows
"(λ n. integral⇧L (μ n) f) ⇢ integral⇧L M f"
proof -
have "0 ≤ B"
using norm_ge_zero f_bdd by (rule order_trans)
note Skorohod
then obtain Omega Y_seq Y where
ps_Omega [simp]: "prob_space Omega" and
Y_seq_measurable [measurable]: "⋀n. Y_seq n ∈ borel_measurable Omega" and
distr_Y_seq: "⋀n. distr Omega borel (Y_seq n) = μ n" and
Y_measurable [measurable]: "Y ∈ borel_measurable Omega" and
distr_Y: "distr Omega borel Y = M" and
YnY: "⋀x :: real. x ∈ space Omega ⟹ (λn. Y_seq n x) ⇢ Y x"  by force
interpret prob_space Omega by fact
have *: "emeasure Omega (Y -` {x. ¬ isCont f x} ∩ space Omega) = 0"
by (subst emeasure_distr [symmetric, where N=borel]) (auto simp: distr_Y discont_null)
have *: "AE x in Omega. (λn. f (Y_seq n x)) ⇢ f (Y x)"
by (rule AE_I [OF _ *]) (auto intro: isCont_tendsto_compose YnY)
show ?thesis
by (auto intro!: integral_dominated_convergence[where w="λx. B"]
simp: f_bdd * integral_distr distr_Y_seq [symmetric] distr_Y [symmetric])
qed

theorem weak_conv_imp_integral_bdd_continuous_conv:
fixes f :: "real ⇒ 'a::{banach, second_countable_topology}"
assumes
"⋀x. isCont f x" and
"⋀x. norm (f x) ≤ B"
shows
"(λ n. integral⇧L (μ n) f) ⇢ integral⇧L M f"
using assms
by (intro weak_conv_imp_bdd_ae_continuous_conv)
(auto intro!: borel_measurable_continuous_onI continuous_at_imp_continuous_on)

theorem weak_conv_imp_continuity_set_conv:
fixes f :: "real ⇒ real"
assumes [measurable]: "A ∈ sets borel" and "M (frontier A) = 0"
shows "(λn. measure (μ n) A) ⇢ measure M A"
proof -
interpret M: real_distribution M by fact
interpret μ: real_distribution "μ n" for n by fact

have "(λn. (∫x. indicator A x ∂μ n) :: real) ⇢ (∫x. indicator A x ∂M)"
by (intro weak_conv_imp_bdd_ae_continuous_conv[where B=1])
(auto intro: assms simp: isCont_indicator)
then show ?thesis
by simp
qed

end

definition
cts_step :: "real ⇒ real ⇒ real ⇒ real"
where
"cts_step a b x ≡ if x ≤ a then 1 else if x ≥ b then 0 else (b - x) / (b - a)"

lemma cts_step_uniformly_continuous:
assumes [arith]: "a < b"
shows "uniformly_continuous_on UNIV (cts_step a b)"
unfolding uniformly_continuous_on_def
proof clarsimp
fix e :: real assume [arith]: "0 < e"
let ?d = "min (e * (b - a)) (b - a)"
have "?d > 0"
by (auto simp add: field_simps)
moreover have "dist x' x < ?d ⟹ dist (cts_step a b x') (cts_step a b x) < e" for x x'
by (auto simp: dist_real_def divide_simps cts_step_def)
ultimately show "∃d > 0. ∀x x'. dist x' x < d ⟶ dist (cts_step a b x') (cts_step a b x) < e"
by blast
qed

lemma (in real_distribution) integrable_cts_step: "a < b ⟹ integrable M (cts_step a b)"
by (rule integrable_const_bound [of _ 1]) (auto simp: cts_step_def[abs_def])

lemma (in real_distribution) cdf_cts_step:
assumes [arith]: "x < y"
shows "cdf M x ≤ integral⇧L M (cts_step x y)" and "integral⇧L M (cts_step x y) ≤ cdf M y"
proof -
have "cdf M x = integral⇧L M (indicator {..x})"
by (simp add: cdf_def)
also have "… ≤ expectation (cts_step x y)"
by (intro integral_mono integrable_cts_step)
(auto simp: cts_step_def less_top[symmetric] split: split_indicator)
finally show "cdf M x ≤ expectation (cts_step x y)" .
next
have "expectation (cts_step x y) ≤ integral⇧L M (indicator {..y})"
by (intro integral_mono integrable_cts_step)
(auto simp: cts_step_def less_top[symmetric] split: split_indicator)
also have "… = cdf M y"
by (simp add: cdf_def)
finally show "expectation (cts_step x y) ≤ cdf M y" .
qed

context
fixes M_seq :: "nat ⇒ real measure"
and M :: "real measure"
assumes distr_M_seq [simp]: "⋀n. real_distribution (M_seq n)"
assumes distr_M [simp]: "real_distribution M"
begin

theorem continuity_set_conv_imp_weak_conv:
fixes f :: "real ⇒ real"
assumes *: "⋀A. A ∈ sets borel ⟹ M (frontier A) = 0 ⟹ (λ n. (measure (M_seq n) A)) ⇢ measure M A"
shows "weak_conv_m M_seq M"
proof -
interpret real_distribution M by simp
show ?thesis
by (auto intro!: * simp: frontier_real_atMost isCont_cdf emeasure_eq_measure weak_conv_m_def weak_conv_def cdf_def2)
qed

theorem integral_cts_step_conv_imp_weak_conv:
assumes integral_conv: "⋀x y. x < y ⟹ (λn. integral⇧L (M_seq n) (cts_step x y)) ⇢ integral⇧L M (cts_step x y)"
shows "weak_conv_m M_seq M"
unfolding weak_conv_m_def weak_conv_def
proof (clarsimp)
interpret real_distribution M by (rule distr_M)
fix x assume "isCont (cdf M) x"
hence left_cont: "continuous (at_left x) (cdf M)"
unfolding continuous_at_split ..
{ fix y :: real assume [arith]: "x < y"
have "limsup (λn. cdf (M_seq n) x) ≤ limsup (λn. integral⇧L (M_seq n) (cts_step x y))"
by (auto intro!: Limsup_mono always_eventually real_distribution.cdf_cts_step)
also have "… = integral⇧L M (cts_step x y)"
by (intro lim_imp_Limsup) (auto intro: integral_conv)
also have "… ≤ cdf M y"
by (simp add: cdf_cts_step)
finally have "limsup (λn. cdf (M_seq n) x) ≤ cdf M y" .
} note * = this
{ fix y :: real assume [arith]: "x > y"
have "cdf M y ≤ ereal (integral⇧L M (cts_step y x))"
by (simp add: cdf_cts_step)
also have "… = liminf (λn. integral⇧L (M_seq n) (cts_step y x))"
by (intro lim_imp_Liminf[symmetric]) (auto intro: integral_conv)
also have "… ≤ liminf (λn. cdf (M_seq n) x)"
by (auto intro!: Liminf_mono always_eventually real_distribution.cdf_cts_step)
finally have "liminf (λn. cdf (M_seq n) x) ≥ cdf M y" .
} note ** = this

have "limsup (λn. cdf (M_seq n) x) ≤ cdf M x"
proof (rule tendsto_lowerbound)
show "∀⇩F i in at_right x. limsup (λxa. ereal (cdf (M_seq xa) x)) ≤ ereal (cdf M i)"
by (subst eventually_at_right[of _ "x + 1"]) (auto simp: * intro: exI [of _ "x+1"])
qed (insert cdf_is_right_cont, auto simp: continuous_within)
moreover have "cdf M x ≤ liminf (λn. cdf (M_seq n) x)"
proof (rule tendsto_upperbound)
show "∀⇩F i in at_left x. ereal (cdf M i) ≤ liminf (λxa. ereal (cdf (M_seq xa) x))"
by (subst eventually_at_left[of "x - 1"]) (auto simp: ** intro: exI [of _ "x-1"])
qed (insert left_cont, auto simp: continuous_within)
ultimately show "(λn. cdf (M_seq n) x) ⇢ cdf M x"
by (elim limsup_le_liminf_real)
qed

theorem integral_bdd_continuous_conv_imp_weak_conv:
assumes
"⋀f. (⋀x. isCont f x) ⟹ (⋀x. abs (f x) ≤ 1) ⟹ (λn. integral⇧L (M_seq n) f::real) ⇢ integral⇧L M f"
shows
"weak_conv_m M_seq M"
apply (rule integral_cts_step_conv_imp_weak_conv [OF assms])
apply (rule continuous_on_interior)
apply (rule uniformly_continuous_imp_continuous)
apply (rule cts_step_uniformly_continuous)
apply (auto simp: cts_step_def)
done

end

end
```