Definition 1.11: Sub-exponential random variables #
A random variable X is sub-exponential with parameter λ (denoted X ~ subE(λ)) if
E[X] = 0 and its moment generating function satisfies:
E[exp(sX)] ≤ exp(s²λ²/2) for all |s| ≤ 1/λ.
This is similar to the sub-Gaussian condition (Definition 1.2) but the MGF bound only holds
in a restricted range |s| ≤ 1/λ rather than for all s ∈ ℝ.
This definition is used in Lemma 1.12 and Theorem 1.13.
def
IsSubExponential
{Ω : Type u_1}
[MeasurableSpace Ω]
{μ : MeasureTheory.Measure Ω}
[MeasureTheory.IsProbabilityMeasure μ]
(X : Ω → ℝ)
(lambda : ℝ)
:
A random variable X on a probability space is sub-exponential with parameter λ
if λ > 0, X is integrable, E[X] = 0, and E[exp(sX)] ≤ exp(s²λ²/2)
for all |s| ≤ 1/λ.