Subgaussian variables and Orlicz norms

For a centered X, call σ² a subgaussian variance proxy if log E exp(λX)≤σ²λ²/2 for every real λ. Chernoff optimization gives a one-sided bound exp(−t²/(2σ²)) and twice that value for two sides. Differentiating near zero shows Var(X)≤σ², but a variable's actual variance need not itself be a valid global proxy. A rare bounded outlier can have small variance while its large-λ mgf needs a larger quadratic envelope.

For independent centered X_i with proxies σ_i², a weighted sum with deterministic coefficients a_i has proxy Σa_i²σ_i². This follows by factoring its mgf and substituting λa_i, including negative coefficients because the hypothesis covers every real λ. Without independence this squared-sum formula is not justified. If all terms are the same random sign, their weighted sum fluctuates at the sum of weights scale rather than at the square-root-of-squares scale. Dependence cannot be repaired by a marginal subgaussian label.

For α equal to one or two, define the ψ_α norm as the infimum of c>0 for which E exp((|X|/c)^α)≤2. The choice two is part of the convention. Changing it changes exact norm values, even though the same distributions retain finite norm. Unlike the centered mgf proxy, this definition applies directly to a noncentered variable and responds to a deterministic offset. A nonzero constant therefore has positive Orlicz norm despite having zero centered variance proxy.

A finite ψ₂ norm is equivalent, up to universal constants, to a Gaussian-shaped two-sided tail and to square-root growth of L^p norms. For a centered variable it is also equivalent to a global quadratic mgf bound, again with constants. A finite ψ₁ norm gives exponential tails and linear moment growth, connecting to the previous lesson's local-mgf subexponential class after centering. These equivalences do not identify the best mgf proxy with the square of the Orlicz norm; computing one quantity does not authorize dropping the conversion factor.

A symmetric sign ε takes values ±1 with equal probabilities. Its mgf is cosh(λ), bounded by exp(λ²/2), and the second derivative at zero makes proxy one optimal. Since ε² is identically one, its ψ₂ norm is one divided by the square root of log two. Both are exact calculations, and the unequal constants already disprove automatic equality of proxy and squared norm. Multiplication by a deterministic scalar scales the norm by its absolute value and the proxy by its square.

For a centered Gaussian of variance τ², the mgf is exactly exp(τ²λ²/2). A separate Gaussian integral gives E exp(G²/c²)=(1−2τ²/c²)^(−1/2) when c²>2τ², with divergence otherwise. Solving the factor-two threshold yields ψ₂ norm sqrt(8/3) times τ. This domain restriction is another instance of the first lesson's rule: the quadratic exponential integral cannot be continued past its finite-moment boundary. It also explains why squaring a Gaussian gives subexponential rather than subgaussian tails.