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A search for double electron capture ($2\varepsilon$), electron capture withpositron emission ($\varepsilon\beta^+$), 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 Sassounderground laboratory. A 627 g sample of cerium oxide deeply purified byliquid-liquid extraction method was used as a source of $\gamma$ quantaexpected in double $\beta$ decay of the cerium isotopes. New improved half-lifelimits were set on different modes and channels of double $\beta$ decay of$^{136}$Ce and $^{138}$Ce at the level of $T_{1/2}>10^{17}-10^{18}$ yr.

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