ProbabilityTheory.Kernel.HasSubexponentialMGF.measure_ge_le
Lemma
From the authors
Bernstein-type tail bound for the upper tail of a sub-exponential random variable.
When V ≤ 0 or b ≤ 0, the right-hand side is at least 1 and the bound is trivial.
Types
-
Ω : Type u_1mΩ : MeasurableSpace ΩA measurable space is a space equipped with a σ-algebra. -
Ω' : Type u_2mΩ' : MeasurableSpace Ω'
Given
-
ν : MeasureTheory.Measure Ω'A measure is defined to be an outer measure that is countably additive on measurable sets, with the additional assumption that the outer measure is the canonical extension of the restricted measure. -
κ : Kernel Ω' ΩA kernel from a measurable spaceαto another measurable spaceβis a measurable functionκ : α → Measure β. -
X : Ω → ℝ -
V : ℝ -
b : ℝ -
t : ℝ
Assuming
Then
∀ᵐ (ω' : Ω') ∂ν, (κ ω').real {ω | t ≤ X ω} ≤ Real.exp (-min (t ^ 2 / (2 * V)) (t / (2 * b)))f.Eventually p or ∀ᶠ x in f, p x mean that {x | p x} ∈ f.MeasurableSpace : Type u_6 → Type u_6A measurable space is a space equipped with a σ-algebra.
MeasureTheory.Measure : (α : Type u_5) → [MeasurableSpace α] → Type u_5A measure is defined to be an outer measure that is countably additive on measurable sets, with the additional assumption that the outer measure is the canonical extension of the restricted measure. The measure of a set `s`, denoted `μ s`, is an extended nonnegative real. The real-valued version is written `μ.real s`.
ProbabilityTheory.Kernel : (α : Type u_1) → (β : Type u_2) → [MeasurableSpace α] → [MeasurableSpace β] → Type (max u_1 u_2)A kernel from a measurable space `α` to another measurable space `β` is a measurable function `κ : α → Measure β`. The measurable space structure on `MeasureTheory.Measure β` is given by `MeasureTheory.Measure.instMeasurableSpace`. A map `κ : α → MeasureTheory.Measure β` is measurable iff `∀ s : Set β, MeasurableSet s → Measurable (fun a ↦ κ a s)`.
Real : TypeThe type `ℝ` of real numbers constructed as equivalence classes of Cauchy sequences of rational numbers.
ProbabilityTheory.Kernel.HasSubexponentialMGF : {Ω : Type u_1} →
{Ω' : Type u_2} →
{mΩ : MeasurableSpace Ω} →
{mΩ' : MeasurableSpace Ω'} →
(Ω → ℝ) →
ℝ →
ℝ →
ProbabilityTheory.Kernel Ω' Ω →
autoParam (MeasureTheory.Measure Ω') ProbabilityTheory.Kernel.HasSubexponentialMGF._auto_1 → PropA random variable `X` has a sub-exponential moment-generating function with parameters `(V, b)` with respect to a kernel `κ` and a measure `ν` if for every `t` with `b * |t| ≤ 1`, `exp (t * X)` is integrable with respect to `κ ∘ₘ ν` and, for `ν`-almost all `ω'`, the moment-generating function of `X` with respect to `κ ω'` is bounded by `exp (V * t ^ 2 / 2)`. For `b = 0` this is `Kernel.HasSubgaussianMGF X V κ ν`.Go to its page
LE.le : {α : Type u} → [self : LE α] → α → α → PropThe less-equal relation: `x ≤ y` Conventions for notations in identifiers: * The recommended spelling of `≤` in identifiers is `le`.
Filter.Eventually : {α : Type u_1} → (α → Prop) → Filter α → Prop`f.Eventually p` or `∀ᶠ x in f, p x` mean that `{x | p x} ∈ f`. E.g., `∀ᶠ x in atTop, p x`
means that `p` holds true for sufficiently large `x`.MeasureTheory.Measure.real : {α : Type u_5} → {m : MeasurableSpace α} → MeasureTheory.Measure α → Set α → ℝThe real-valued version of a measure. Maps infinite measure sets to zero. Use as `μ.real s`. The API is developed in `Mathlib/MeasureTheory/Measure/Real.lean`.
Set.ofPred : {α : Type u} → (α → Prop) → Set αTurn a predicate `p : α → Prop` into a set, also written as `{x | p x}`Real.exp : ℝ → ℝThe real exponential function, defined as the real part of the complex exponential
Neg.neg : {α : Type u} → [self : Neg α] → α → α`-a` computes the negative or opposite of `a`. The meaning of this notation is type-dependent. Conventions for notations in identifiers: * The recommended spelling of `-` in identifiers is `neg` (when used as a unary operator).
Min.min : {α : Type u} → [self : Min α] → α → α → αReturns the lesser of its two arguments. Conventions for notations in identifiers: * The recommended spelling of `min` in identifiers is `min`. * The recommended spelling of `⊓` in identifiers is `inf` (`⊓` is the preferred notation for `min` when the type is not linearly ordered.).
HPow.hPow : {α : Type u} → {β : Type v} → {γ : outParam (Type w)} → [self : HPow α β γ] → α → β → γ`a ^ b` computes `a` to the power of `b`. The meaning of this notation is type-dependent. Conventions for notations in identifiers: * The recommended spelling of `^` in identifiers is `pow`.
HDiv.hDiv : {α : Type u} → {β : Type v} → {γ : outParam (Type w)} → [self : HDiv α β γ] → α → β → γ`a / b` computes the result of dividing `a` by `b`. The meaning of this notation is type-dependent. * For most types like `Nat`, `Int`, `Rat`, `Real`, `a / 0` is defined to be `0`. * For `Nat`, `a / b` rounds downwards. * For `Int`, `a / b` rounds downwards if `b` is positive or upwards if `b` is negative. It is implemented as `Int.ediv`, the unique function satisfying `a % b + b * (a / b) = a` and `0 ≤ a % b < natAbs b` for `b ≠ 0`. Other rounding conventions are available using the functions `Int.fdiv` (floor rounding) and `Int.tdiv` (truncation rounding). * For `Float`, `a / 0` follows the IEEE 754 semantics for division, usually resulting in `inf` or `nan`. Conventions for notations in identifiers: * The recommended spelling of `/` in identifiers is `div`.
HMul.hMul : {α : Type u} → {β : Type v} → {γ : outParam (Type w)} → [self : HMul α β γ] → α → β → γ`a * b` computes the product of `a` and `b`. The meaning of this notation is type-dependent. Conventions for notations in identifiers: * The recommended spelling of `*` in identifiers is `mul`.
Code
lemma measure_ge_le (h : HasSubexponentialMGF X V b κ ν) {t : ℝ} (ht : 0 ≤ t) :
∀ᵐ ω' ∂ν, (κ ω').real {ω | t ≤ X ω} ≤ exp (-min (t ^ 2 / (2 * V)) (t / (2 * b)))Proof
by
rcases le_or_gt b 0 with hb | hb
· filter_upwards [h.measureReal_le_one _] with ω' h
refine h.trans (one_le_exp ?_)
rw [neg_nonneg]
exact (min_le_right _ _).trans (div_nonpos_of_nonneg_of_nonpos ht (by linarith))
rcases le_or_gt (t * b) V with htb | htb
· filter_upwards [h.measure_ge_le_exp_neg_sq ht htb] with ω' h
exact h.trans (exp_le_exp.2 (neg_le_neg (min_le_left _ _)))
· filter_upwards [h.measure_ge_le_exp_neg_div hb htb.le] with ω' h
exact h.trans (exp_le_exp.2 (neg_le_neg (min_le_right _ _)))Meaning last changed in v4.34.0-rc2-76-g565f652 (2026-09-10).
Self-contained, with its dependencies inlined and proofs replaced by sorry: download the raw file · open it in the Lean web editor.
Dependency graph
Audit surface: 1 project declarations, 68 external constants
✓ Proved: no sorry anywhere in its closure
This is the tool's own reading of one build's recorded axioms, and it is not robust against an author who wants it to pass. Checking meant to be relied on should go through Comparator, which replays the proof through the kernel from an export against an explicit list of permitted axioms.