Multivariate analog of the Le Roy-Lindelöf theorem about power series analytic continuation

We consider the problem of continuability into a sectorial domain for multiple power series and prove the multivariate analog of the Le Roy-Lindelöf theorem, i.e. establish a connection between the growth of the holomorphic function interpolating the series coefficients in the imaginary subspace and the sectoral domain where the multiple power series is analytically continued to.