  Corentin Perret-Gentil proved, under some very general conditions, that short
sums of $\ell$-adic trace functions over finite fields of varying center
converges in law to a Gaussian random variable or vector. The main inputs are
P.~Deligne's equidistribution theorem, N.~Katz' works and the results surveyed
in \cite{MR3338119}. In particular, this applies to $2$-dimensional Kloosterman
sums $\mathsf{Kl}_{2,\mathbb{F}_q}$ studied by N.~Katz in \cite{MR955052} and
in \cite{MR1081536} when the field $\mathbb{F}_q$ gets large. \par This article
considers the case of short sums of normalized classical Kloosterman sums of
prime powers moduli $\mathsf{Kl}_{p^n}$, as $p$ tends to infinity among the
prime numbers and $n\geq 2$ is a fixed integer. A convergence in law towards a
real-valued standard Gaussian random variable is proved under some very natural
conditions.
