I'd like to replace the two Orlicz families in
Moments/Orlicz.lean
with a single ψ_p family, and use it to tidy up the sub-Gaussian side. Mathlib
has no Orlicz norm at the current pin.
Why:
- Lines 57–342 (
ψ₂) and 343–588 (ψ₁) are a near line-by-line transcription of
each other: 3 definitions and 13 lemmas duplicated. SubGaussianOrlicz.lean
repeats three more pairs.
- Nothing relates the two norms, so "sub-Gaussian implies sub-exponential"
cannot be stated today.
‖c‖_{ψ₂} = |c| / √(log 2) and ‖c‖_{ψ₁} = |c| / log 2 are both
|c| / (log 2) ^ (1 / p), which only becomes visible with p a parameter.
Proposed definition:
/-- `K` is an admissible `ψ p` scale for `X`: `E[exp ((|X| / K) ^ p)] ≤ 2`. -/
def HasOrliczPsiBound (p : ℝ) (X : Ω → ℝ) (μ : Measure Ω) (K : ℝ) : Prop :=
0 < K ∧ ∫⁻ ω, ENNReal.ofReal (exp ((|X ω| / K) ^ p)) ∂μ ≤ 2
plus orliczPsiNorm p and HasFiniteOrliczPsiNorm p.
I'd like to replace the two Orlicz families in
Moments/Orlicz.leanwith a single
ψ_pfamily, and use it to tidy up the sub-Gaussian side. Mathlibhas no Orlicz norm at the current pin.
Why:
ψ₂) and 343–588 (ψ₁) are a near line-by-line transcription ofeach other: 3 definitions and 13 lemmas duplicated.
SubGaussianOrlicz.leanrepeats three more pairs.
cannot be stated today.
‖c‖_{ψ₂} = |c| / √(log 2)and‖c‖_{ψ₁} = |c| / log 2are both|c| / (log 2) ^ (1 / p), which only becomes visible withpa parameter.Proposed definition:
plus
orliczPsiNorm pandHasFiniteOrliczPsiNorm p.