Double-beta processes in $^{184}$Os and $^{192}$Os were searched for over 15851 h at the Gran Sasso National Laboratory (LNGS) of the I.N.F.N. by using a 118 g ultra-pure osmium sample installed on the endcap of a 112 cm$^3$ ultra-low-background broad-energy germanium detector. New limits on double-electron capture and electron capture with positron emission in $^{184}$Os were set at the level of $lim T_{1/2} sim 10^{16}-10^{17}$ yr. In particular the $2 u$2K and $2 u$KL decays of $^{184}$Os to the ground state of $^{184}$W are restricted as $T_{1/2}geq3.0times 10^{16}$ yr and $T_{1/2}geq2.0times 10^{16}$ yr, respectively. A lower limit on the half-life for the double-beta decay of $^{192}$Os to the first excited level of $^{192}$Pt was set as $lim T_{1/2}=2.0times 10^{20}$ yr at 90% C.L.
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