  Building on the mapping of large-$S$ spin chains onto the O($3$) nonlinear
$\sigma$ model with coupling constant $2/S$, and on general properties of that
model (asymptotic freedom, implying that perturbation theory is valid at high
energy, and Elitzur's conjecture that rotationally invariant quantities are
infrared finite in perturbation theory), we use the Holstein-Primakoff
representation to derive analytic expressions for the equal-time and dynamical
spin-spin correlations valid at distances smaller than $S^{-1} \exp(\pi S)$ or
at energies larger than $J S^2 \exp(-\pi S)$, where $J$ is the Heisenberg
exchange coupling. This is supported by comparing the static correlations with
quantum Monte Carlo simulations for $S = 5/2$.
