  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\nu$2K and $2\nu$KL decays of $^{184}$Os to the ground state
of $^{184}$W are restricted as $T_{1/2}\geq3.0\times 10^{16}$ yr and
$T_{1/2}\geq2.0\times 10^{16}$ 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 $\lim T_{1/2}=2.0\times 10^{20}$ yr at 90\% C.L.
