the variance proxy
The variance proxy is the parameter that quantifies how sub-Gaussian a variable is, and it is the single number that propagates through every Hoeffding-style bound. For a centered random variable X, we say X is sub-Gaussian with variance proxy sigma^2 if E[exp(lambda X)] <= exp(sigma^2 lambda^2 / 2) for all real lambda. The name is deliberate: a genuine Gaussian N(0, sigma^2) achieves this with equality, so sigma^2 plays the role the true variance plays for a Gaussian, even when X is far from Gaussian.
The proxy is an upper bound on the actual variance, never below it: Var(X) <= sigma^2, because differentiating the MGF inequality twice at lambda = 0 gives E[X^2] <= sigma^2. But the proxy can be strictly larger than the variance. A Rademacher sign has variance 1 and proxy exactly 1 (a tight match), but a variable concentrated near +/-1 with a tiny mass elsewhere can have variance just under 1 while its proxy is forced up by the rare large values that dominate the MGF for large lambda. The proxy is what the tail sees; the variance is only what the second moment sees. For an [a,b]-bounded variable, Hoeffding's lemma gives the proxy (b-a)^2/4, which can be much larger than the true variance — this gap is the price Hoeffding pays for ignoring the variance, and it is exactly what Bernstein and Bennett recover.
The reason the proxy matters is additivity. For independent centered sub-Gaussians X_1, ..., X_n with proxies sigma_1^2, ..., sigma_n^2, the MGFs multiply, so the sum has proxy sigma_1^2 + ... + sigma_n^2 — exactly as variances of independent variables add. Feeding this summed proxy into the Chernoff bound yields the general Hoeffding inequality P(sum >= t) <= exp(-t^2 / (2 sum sigma_i^2)). The whole logic of sub-Gaussian concentration is: identify a variance proxy, add them up, optimize over lambda. The honest caveat is that the proxy is the quantity controlling deviations, and when it overestimates the variance your tail bound is loose; the sharper inequalities (Bernstein, Bennett, the entropy method) exist precisely to use the true variance instead of a possibly-pessimistic proxy.
Take a coin that is 1 with probability p and -1 with probability 1-p (so not centered); after centering, X = sign - (2p-1). Its variance is 4p(1-p), which is tiny when p is near 0 or 1, yet for a bounded variable Hoeffding still uses proxy 1 (range = 2, so (b-a)^2/4 = 1). The proxy ignores that a very biased coin barely fluctuates — which is exactly the slack Bernstein's variance-aware bound removes.
The proxy upper-bounds the variance and can be far larger when the variable is highly biased.
Always Var(X) <= sigma^2, never the reverse; the optimal (smallest) proxy is what makes a bound tight. A common error is to call sigma^2 'the variance' — it is a proxy, and the looseness when it overshoots the true variance is exactly why one upgrades from Hoeffding to Bernstein.