  We carried out the comparative study of the signal from the decay of double
$K$-shell vacancy production that follows after single $K$-shell electron
capture of $^{81}$Kr and double $K$-shell electron capture of $^{78}$Kr. The
radiative decay of a the double $1s$ vacancy state was identified by detecting
the triple coincidence of two $K$ X-rays and several Auger electrons in the
$ECEC$-decay, or by detecting two $K$ X-rays and (Auger electrons + ejected
$K$-shell electron) in the $EC$ decay. The number of $K$-shell vacancies per
the $K$-electron capture, produced as a result of the shake-off process, has
been measured for the decay of $^{81}$Kr. The probability for this decay was
found to be $P_{KK}=(5.7\pm0.8)\times10^{-5}$ with a systematic error of
$(\Delta P_{KK})_{syst}=\pm0.4 \times10^{-5}$. For the $^{78}{\rm{Kr}}(2\nu2K)$
decay, the comparative study of single- and double-capture decays allowed us to
obtain the signal-to-background ratio to be 15/1. The half-life
$T_{1/2}^{2\nu2K}(g.s. \rightarrow g.s.) =
[1.9^{+1.3}_{-0.7}(stat)\pm0.3(syst)]\times 10^{22}$ y is determined from the
analysis of data that have been accumulated over 782 days of live measurements
in the experiment that used samples consisted of 170.6 g of $^{78}$Kr.
