  How heterogeneous multiscale methods (HMM) handle fluctuations acting on the
slow variables in fast-slow systems is investigated. In particular, it is shown
via analysis of central limit theorems (CLT) and large deviation principles
(LDP) that the standard version of HMM artificially amplifies these
fluctuations. A simple modification of HMM, termed parallel HMM, is introduced
and is shown to remedy this problem, capturing fluctuations correctly both at
the level of the CLT and the LDP. Similar type of arguments can also be used to
justify that the tau-leaping method used in the context of Gillespie's
stochastic simulation algorithm for Markov jump processes also captures the
right CLT and LDP for these processes.
