= Conditional concentration at a large-deviation threshold
{title2=$\mathbb P(S_n/n\in[a,a+\varepsilon)\mid S_n/n\geq a)\to1$}
For independent identically distributed real variables with everywhere finite <cumulant-generating function>, let $\mu$ be the mean. If $a\geq\mu$ and $\mathbb P(X_1>a)>0$, then $\mathbb P(S_n/n\in[a,a+\varepsilon)\mid S_n/n\geq a)\to1$ for each $\varepsilon>0$. The <Legendre transform of a cumulant-generating function> is finite at $a$, and a positive exponential parameter gives strictly greater cost at $a+\varepsilon$. A <Chernoff bound> for the farther tail divided by the <Cramér theorem> lower estimate at $a$ decays exponentially. The support condition both ensures positive conditioning probabilities and avoids a vacuous conditioning event.
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