  The Monte Carlo (MC) estimates of thermal averages are usually functions of
system control parameters $\lambda $, such as temperature, volume, interaction
couplings, etc. Given the MC average at a set of prescribed control parameters
$\lambda_{0}$, the problem of analytic continuation of the MC data to $\lambda
$-values in the neighborhood of $\lambda_{0}$ is considered in both classic and
quantum domains. The key result is the theorem that links the differential
properties of thermal averages to the higher-order cumulants. The theorem and
analytic continuation formulas expressed via higher-order cumulants are
numerically tested on the classical Lennard-Jones cluster system of N=13, 55,
and 147 neon particles.
