  In this letter, we investigate the statistical properties of electromagnetic
signals after different times of duration within one-dimensional
local-disordered time-varying cavities, where both spatial and temporal
disorders are added. Our findings reveal that, in the vast majority of cases,
adequate temporal disorder in local space can make the electromagnetic field
statistically localized, obeying a normal distribution at a specific point in
time of arbitrary location within the cavity. We employ the concept of
disordered space-time crystals and leverage Lindeberg's and Lyapunov's theorems
to theoretically prove the normal distribution of the field values.
Furthermore, we find that with the increase of energy provided by time
variation, the probability of extreme fields will significantly increase and
the field intensity eventually is de-normalized, that is, deviating from the
normal distribution. This study not only sheds light on the statistical
properties of transient signals in local-disordered time-varying systems but
also paves the way for further exploration in wave dynamics of analogous
systems.
