New limits on $2\epsilon$, $\epsilon {\beta }^{+}$ and $2{\beta }^{+}$ decay of ${}^{136}$Ce and ${}^{138}$Ce with deeply purified cerium sample

A search for double electron capture ($2\epsilon$), electron capture with positron emission ($\epsilon {\beta }^{+}$), and double positron emission $2{\beta }^{+}$) in ${}^{136}$Ce and ${}^{138}$Ce was realized with a 465 cm${}^3$ ultra-low background HP Ge $\gamma$ spectrometer over 2299 h at the Gran Sasso underground laboratory. A 627 g sample of cerium oxide deeply purified by liquid-liquid extraction method was used as a source of $\gamma$ quanta expected in double $\beta$ decay of the cerium isotopes. New improved half-life limits were set on different modes and channels of double $\beta$ decay of ${}^{136}$Ce and ${}^{138}$Ce at the level of $T_{1\mathrm{/}2}>{10}^{17}-{10}^{18}$ yr.