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We carried out the comparative study of the signal from the decay of double$K$-shell vacancy production that follows after single $K$-shell electroncapture of $^{81}$Kr and double $K$-shell electron capture of $^{78}$Kr. Theradiative decay of a the double $1s$ vacancy state was identified by detectingthe 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 perthe $K$-electron capture, produced as a result of the shake-off process, hasbeen measured for the decay of $^{81}$Kr. The probability for this decay wasfound 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 toobtain 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 theanalysis of data that have been accumulated over 782 days of live measurementsin the experiment that used samples consisted of 170.6 g of $^{78}$Kr.

Comparative study of the double $K$-shell-vacancy production in single- and double-electron capture decay