  We study the evolution leading to (or regressing from) a large fluctuation in
a Statistical Mechanical system. We introduce and study analytically a simple
model of many identically and independently distributed microscopic variables
$n_m$ ($m=1,M$) evolving by means of a master equation. We show that the
process producing a non-typical fluctuation with a value of $N=\sum_{m=1}^Mn_m$
well above the average $\langle N\rangle$ is slow. Such process is
characterized by the power-law growth of the largest possible observable value
of $N$ at a given time $t$. We find similar features also for the reverse
process of the regression from a rare state with $N\gg \langle N\rangle$ to a
typical one with $N \simeq \langle N\rangle$.
