Properties of Sub-Gaussian Moments and MGFs

SECTION 01Properties of Sub-Gaussian Moments and MGFs

We introduce some additional properties of sub-Gaussian moments that will be useful later.

Claim · MGF Control and Moment Control Are Equivalent

Let X have mean zero. The following two conditions are equivalent:

(1) There is a σ > 0 such that

E [e^{λ X}] \leq e^{\frac{λ^{2} σ^{2}}{2}}, λ \in R

(2) There is a θ > 0 such that

E [X^{2 k}] \leq 2 {(2 θ^{2})}^{k} k!, k = 1, 2, \dots

Proof · MGF Control Implies Moment Control

Although differentiating the MGF at zero recovers moments, an inequality between functions does not imply the same inequality between their derivatives. We instead pass through tail probabilities.

The MGF bound gives the sub-Gaussian tail bound

P (| X | \geq t) \leq 2 e^{- \frac{t^{2}}{2 σ^{2}}}, t \geq 0

Relate the moment to the tail using

E [X^{2 k}] = \int_{0}^{\infty} 2 k t^{2 k - 1} P (| X | \geq t) d t

Substitute the tail bound and integrate:

E [X^{2 k}] \leq 4 k \int_{0}^{\infty} t^{2 k - 1} e^{- \frac{t^{2}}{2 σ^{2}}} d t = 2 {(2 σ^{2})}^{k} k!

SECTION 02Gaussian Comparison

Claim · MGF, Gaussian Tail Comparison, and Moment Control Are Equivalent

Let X have mean zero. The following three conditions are equivalent:

(1) There is a σ > 0 such that

E [e^{λ X}] \leq e^{\frac{λ^{2} σ^{2}}{2}}, λ \in R

(2) There is a constant c ≥ 1 and a Gaussian variable Z ∼ N(0, τ²), with τ > 0, such that

P (| X | \geq s) \leq c P (| Z | \geq s), s \geq 0

(3) There is a θ > 0 such that

E [X^{2 k}] \leq \frac{(2 k)!}{2^{k} k!} θ^{2 k}, k = 1, 2, \dots

All three conditions characterize sub-Gaussianity. The parameters may change when passing between the conditions.

Proof · Gaussian Tail Comparison Implies Moment Control

Applying the same tail-integral identity gives

E [X^{2 k}] \leq c E [Z^{2 k}] = c \frac{(2 k)!}{2^{k} k!} τ^{2 k}

The constant c can be absorbed into the moment parameter by taking θ = τ√c.

SECTION 03Connecting the Two Moment Bounds

Sections 1 and 2 describe the same type of moment control, with different constants. For k ≥ 1, we have 2(k!)² ≤ (2k)!, so the bound from Section 1 gives

E [X^{2 k}] \leq 2 {(2 σ^{2})}^{k} k! \leq \frac{(2 k)!}{2^{k} k!} {(2 σ)}^{2 k}

Taking θ = 2σ puts this bound in the Gaussian moment form used in Section 2. The parameters may differ, but both forms characterize sub-Gaussian moment growth.