Identically distributed random variables #
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Two random variables defined on two (possibly different) probability spaces but taking value in the same space are identically distributed if their distributions (i.e., the image probability measures on the target space) coincide. We define this concept and establish its basic properties in this file.
Main definitions and results #
ident_distrib f g μ νregisters that the image ofμunderfcoincides with the image ofνunderg(and thatfandgare almost everywhere measurable, as otherwise the image measures don't make sense). The measures can be kept implicit as inident_distrib f gif the spaces are registered as measure spaces.ident_distrib.comp: being identically distributed is stable under composition with measurable maps.
There are two main kind of lemmas, under the assumption that f and g are identically
distributed: lemmas saying that two quantities computed for f and g are the same, and lemmas
saying that if f has some property then g also has it. The first kind is registered as
ident_distrib.foo_eq, the second one as ident_distrib.foo_snd (in the latter case, to deduce
a property of f from one of g, use h.symm.foo_snd where h : ident_distrib f g μ ν). For
instance:
-
ident_distrib.measure_mem_eq: iffandgare identically distributed, then the probabilities that they belong to a given measurable set are the same. -
ident_distrib.integral_eq: iffandgare identically distributed, then their integrals are the same. -
ident_distrib.variance_eq: iffandgare identically distributed, then their variances are the same. -
ident_distrib.ae_strongly_measurable_snd: iffandgare identically distributed andfis almost everywhere strongly measurable, then so isg. -
ident_distrib.mem_ℒp_snd: iffandgare identically distributed andfbelongs toℒp, then so doesg.
We also register several dot notation shortcuts for convenience.
For instance, if h : ident_distrib f g μ ν, then h.sq states that f^2 and g^2 are
identically distributed, and h.norm states that ‖f‖ and ‖g‖ are identically distributed, and
so on.
structure
probability_theory.ident_distrib
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
(f : α → γ)
(g : β → γ)
(μ : measure_theory.measure α . "volume_tac")
(ν : measure_theory.measure β . "volume_tac") :
Prop
- ae_measurable_fst : ae_measurable f μ
- ae_measurable_snd : ae_measurable g ν
- map_eq : measure_theory.measure.map f μ = measure_theory.measure.map g ν
Two functions defined on two (possibly different) measure spaces are identically distributed if their image measures coincide. This only makes sense when the functions are ae measurable (as otherwise the image measures are not defined), so we require this as well in the definition.
@[protected]
theorem
probability_theory.ident_distrib.refl
{α : Type u_1}
{γ : Type u_3}
[measurable_space α]
[measurable_space γ]
{μ : measure_theory.measure α}
{f : α → γ}
(hf : ae_measurable f μ) :
probability_theory.ident_distrib f f μ μ
@[protected]
theorem
probability_theory.ident_distrib.symm
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
{f : α → γ}
{g : β → γ}
(h : probability_theory.ident_distrib f g μ ν) :
probability_theory.ident_distrib g f ν μ
@[protected]
theorem
probability_theory.ident_distrib.trans
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
{δ : Type u_4}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
[measurable_space δ]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
{f : α → γ}
{g : β → γ}
{ρ : measure_theory.measure δ}
{h : δ → γ}
(h₁ : probability_theory.ident_distrib f g μ ν)
(h₂ : probability_theory.ident_distrib g h ν ρ) :
probability_theory.ident_distrib f h μ ρ
@[protected]
theorem
probability_theory.ident_distrib.comp_of_ae_measurable
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
{δ : Type u_4}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
[measurable_space δ]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
{f : α → γ}
{g : β → γ}
{u : γ → δ}
(h : probability_theory.ident_distrib f g μ ν)
(hu : ae_measurable u (measure_theory.measure.map f μ)) :
probability_theory.ident_distrib (u ∘ f) (u ∘ g) μ ν
@[protected]
theorem
probability_theory.ident_distrib.comp
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
{δ : Type u_4}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
[measurable_space δ]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
{f : α → γ}
{g : β → γ}
{u : γ → δ}
(h : probability_theory.ident_distrib f g μ ν)
(hu : measurable u) :
probability_theory.ident_distrib (u ∘ f) (u ∘ g) μ ν
@[protected]
theorem
probability_theory.ident_distrib.of_ae_eq
{α : Type u_1}
{γ : Type u_3}
[measurable_space α]
[measurable_space γ]
{μ : measure_theory.measure α}
{f g : α → γ}
(hf : ae_measurable f μ)
(heq : f =ᵐ[μ] g) :
probability_theory.ident_distrib f g μ μ
theorem
probability_theory.ident_distrib.measure_mem_eq
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
{f : α → γ}
{g : β → γ}
(h : probability_theory.ident_distrib f g μ ν)
{s : set γ}
(hs : measurable_set s) :
theorem
probability_theory.ident_distrib.measure_preimage_eq
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
{f : α → γ}
{g : β → γ}
(h : probability_theory.ident_distrib f g μ ν)
{s : set γ}
(hs : measurable_set s) :
theorem
probability_theory.ident_distrib.ae_snd
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
{f : α → γ}
{g : β → γ}
(h : probability_theory.ident_distrib f g μ ν)
{p : γ → Prop}
(pmeas : measurable_set {x : γ | p x})
(hp : ∀ᵐ (x : α) ∂μ, p (f x)) :
theorem
probability_theory.ident_distrib.ae_mem_snd
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
{f : α → γ}
{g : β → γ}
(h : probability_theory.ident_distrib f g μ ν)
{t : set γ}
(tmeas : measurable_set t)
(ht : ∀ᵐ (x : α) ∂μ, f x ∈ t) :
theorem
probability_theory.ident_distrib.ae_strongly_measurable_fst
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
{f : α → γ}
{g : β → γ}
[topological_space γ]
[topological_space.metrizable_space γ]
[opens_measurable_space γ]
[topological_space.second_countable_topology γ]
(h : probability_theory.ident_distrib f g μ ν) :
In a second countable topology, the first function in an identically distributed pair is a.e.
strongly measurable. So is the second function, but use h.symm.ae_strongly_measurable_fst as
h.ae_strongly_measurable_snd has a different meaning.
theorem
probability_theory.ident_distrib.ae_strongly_measurable_snd
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
{f : α → γ}
{g : β → γ}
[topological_space γ]
[topological_space.metrizable_space γ]
[borel_space γ]
(h : probability_theory.ident_distrib f g μ ν)
(hf : measure_theory.ae_strongly_measurable f μ) :
If f and g are identically distributed and f is a.e. strongly measurable, so is g.
theorem
probability_theory.ident_distrib.ae_strongly_measurable_iff
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
{f : α → γ}
{g : β → γ}
[topological_space γ]
[topological_space.metrizable_space γ]
[borel_space γ]
(h : probability_theory.ident_distrib f g μ ν) :
theorem
probability_theory.ident_distrib.ess_sup_eq
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
{f : α → γ}
{g : β → γ}
[conditionally_complete_linear_order γ]
[topological_space γ]
[opens_measurable_space γ]
[order_closed_topology γ]
(h : probability_theory.ident_distrib f g μ ν) :
theorem
probability_theory.ident_distrib.lintegral_eq
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[measurable_space β]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
{f : α → ennreal}
{g : β → ennreal}
(h : probability_theory.ident_distrib f g μ ν) :
theorem
probability_theory.ident_distrib.integral_eq
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
{f : α → γ}
{g : β → γ}
[normed_add_comm_group γ]
[normed_space ℝ γ]
[complete_space γ]
[borel_space γ]
(h : probability_theory.ident_distrib f g μ ν) :
theorem
probability_theory.ident_distrib.snorm_eq
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
{f : α → γ}
{g : β → γ}
[normed_add_comm_group γ]
[opens_measurable_space γ]
(h : probability_theory.ident_distrib f g μ ν)
(p : ennreal) :
measure_theory.snorm f p μ = measure_theory.snorm g p ν
theorem
probability_theory.ident_distrib.mem_ℒp_snd
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
{f : α → γ}
{g : β → γ}
[normed_add_comm_group γ]
[borel_space γ]
{p : ennreal}
(h : probability_theory.ident_distrib f g μ ν)
(hf : measure_theory.mem_ℒp f p μ) :
measure_theory.mem_ℒp g p ν
theorem
probability_theory.ident_distrib.mem_ℒp_iff
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
{f : α → γ}
{g : β → γ}
[normed_add_comm_group γ]
[borel_space γ]
{p : ennreal}
(h : probability_theory.ident_distrib f g μ ν) :
measure_theory.mem_ℒp f p μ ↔ measure_theory.mem_ℒp g p ν
theorem
probability_theory.ident_distrib.integrable_snd
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
{f : α → γ}
{g : β → γ}
[normed_add_comm_group γ]
[borel_space γ]
(h : probability_theory.ident_distrib f g μ ν)
(hf : measure_theory.integrable f μ) :
theorem
probability_theory.ident_distrib.integrable_iff
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
{f : α → γ}
{g : β → γ}
[normed_add_comm_group γ]
[borel_space γ]
(h : probability_theory.ident_distrib f g μ ν) :
@[protected]
theorem
probability_theory.ident_distrib.norm
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
{f : α → γ}
{g : β → γ}
[normed_add_comm_group γ]
[borel_space γ]
(h : probability_theory.ident_distrib f g μ ν) :
probability_theory.ident_distrib (λ (x : α), ‖f x‖) (λ (x : β), ‖g x‖) μ ν
@[protected]
theorem
probability_theory.ident_distrib.nnnorm
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
{f : α → γ}
{g : β → γ}
[normed_add_comm_group γ]
[borel_space γ]
(h : probability_theory.ident_distrib f g μ ν) :
probability_theory.ident_distrib (λ (x : α), ‖f x‖₊) (λ (x : β), ‖g x‖₊) μ ν
@[protected]
theorem
probability_theory.ident_distrib.pow
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
{f : α → γ}
{g : β → γ}
[has_pow γ ℕ]
[has_measurable_pow γ ℕ]
(h : probability_theory.ident_distrib f g μ ν)
{n : ℕ} :
probability_theory.ident_distrib (λ (x : α), f x ^ n) (λ (x : β), g x ^ n) μ ν
@[protected]
theorem
probability_theory.ident_distrib.sq
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
{f : α → γ}
{g : β → γ}
[has_pow γ ℕ]
[has_measurable_pow γ ℕ]
(h : probability_theory.ident_distrib f g μ ν) :
probability_theory.ident_distrib (λ (x : α), f x ^ 2) (λ (x : β), g x ^ 2) μ ν
@[protected]
theorem
probability_theory.ident_distrib.coe_nnreal_ennreal
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[measurable_space β]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
{f : α → nnreal}
{g : β → nnreal}
(h : probability_theory.ident_distrib f g μ ν) :
probability_theory.ident_distrib (λ (x : α), ↑(f x)) (λ (x : β), ↑(g x)) μ ν
theorem
probability_theory.ident_distrib.mul_const
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
{f : α → γ}
{g : β → γ}
[has_mul γ]
[has_measurable_mul γ]
(h : probability_theory.ident_distrib f g μ ν)
(c : γ) :
probability_theory.ident_distrib (λ (x : α), f x * c) (λ (x : β), g x * c) μ ν
theorem
probability_theory.ident_distrib.add_const
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
{f : α → γ}
{g : β → γ}
[has_add γ]
[has_measurable_add γ]
(h : probability_theory.ident_distrib f g μ ν)
(c : γ) :
probability_theory.ident_distrib (λ (x : α), f x + c) (λ (x : β), g x + c) μ ν
theorem
probability_theory.ident_distrib.const_mul
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
{f : α → γ}
{g : β → γ}
[has_mul γ]
[has_measurable_mul γ]
(h : probability_theory.ident_distrib f g μ ν)
(c : γ) :
probability_theory.ident_distrib (λ (x : α), c * f x) (λ (x : β), c * g x) μ ν
theorem
probability_theory.ident_distrib.const_add
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
{f : α → γ}
{g : β → γ}
[has_add γ]
[has_measurable_add γ]
(h : probability_theory.ident_distrib f g μ ν)
(c : γ) :
probability_theory.ident_distrib (λ (x : α), c + f x) (λ (x : β), c + g x) μ ν
theorem
probability_theory.ident_distrib.sub_const
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
{f : α → γ}
{g : β → γ}
[has_sub γ]
[has_measurable_sub γ]
(h : probability_theory.ident_distrib f g μ ν)
(c : γ) :
probability_theory.ident_distrib (λ (x : α), f x - c) (λ (x : β), g x - c) μ ν
theorem
probability_theory.ident_distrib.div_const
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
{f : α → γ}
{g : β → γ}
[has_div γ]
[has_measurable_div γ]
(h : probability_theory.ident_distrib f g μ ν)
(c : γ) :
probability_theory.ident_distrib (λ (x : α), f x / c) (λ (x : β), g x / c) μ ν
theorem
probability_theory.ident_distrib.const_div
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
{f : α → γ}
{g : β → γ}
[has_div γ]
[has_measurable_div γ]
(h : probability_theory.ident_distrib f g μ ν)
(c : γ) :
probability_theory.ident_distrib (λ (x : α), c / f x) (λ (x : β), c / g x) μ ν
theorem
probability_theory.ident_distrib.const_sub
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
{f : α → γ}
{g : β → γ}
[has_sub γ]
[has_measurable_sub γ]
(h : probability_theory.ident_distrib f g μ ν)
(c : γ) :
probability_theory.ident_distrib (λ (x : α), c - f x) (λ (x : β), c - g x) μ ν
theorem
probability_theory.ident_distrib.evariance_eq
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[measurable_space β]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
{f : α → ℝ}
{g : β → ℝ}
(h : probability_theory.ident_distrib f g μ ν) :
theorem
probability_theory.ident_distrib.variance_eq
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[measurable_space β]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
{f : α → ℝ}
{g : β → ℝ}
(h : probability_theory.ident_distrib f g μ ν) :
theorem
probability_theory.mem_ℒp.uniform_integrable_of_ident_distrib_aux
{α : Type u_1}
[measurable_space α]
{E : Type u_5}
[measurable_space E]
[normed_add_comm_group E]
[borel_space E]
[topological_space.second_countable_topology E]
{μ : measure_theory.measure α}
[measure_theory.is_finite_measure μ]
{ι : Type u_2}
{f : ι → α → E}
{j : ι}
{p : ennreal}
(hp : 1 ≤ p)
(hp' : p ≠ ⊤)
(hℒp : measure_theory.mem_ℒp (f j) p μ)
(hfmeas : ∀ (i : ι), measure_theory.strongly_measurable (f i))
(hf : ∀ (i : ι), probability_theory.ident_distrib (f i) (f j) μ μ) :
This lemma is superceded by mem_ℒp.uniform_integrable_of_ident_distrib which only require
ae_strongly_measurable.
theorem
probability_theory.mem_ℒp.uniform_integrable_of_ident_distrib
{α : Type u_1}
[measurable_space α]
{E : Type u_5}
[measurable_space E]
[normed_add_comm_group E]
[borel_space E]
[topological_space.second_countable_topology E]
{μ : measure_theory.measure α}
[measure_theory.is_finite_measure μ]
{ι : Type u_2}
{f : ι → α → E}
{j : ι}
{p : ennreal}
(hp : 1 ≤ p)
(hp' : p ≠ ⊤)
(hℒp : measure_theory.mem_ℒp (f j) p μ)
(hf : ∀ (i : ι), probability_theory.ident_distrib (f i) (f j) μ μ) :
A sequence of identically distributed Lᵖ functions is p-uniformly integrable.