Conditions for recurrence and transience for time-inhomogeneous random walks

The present paper extends the earlier results obtained by Abramov [Conditions for recurrence and transience for time-inhomogeneous birth-and-death processes' \emph{Bull. Aust. Math. Soc.} \textbf{109} (2024), 393--402] for the case of time-inhomogeneous random walks, the increments of which take values in $\mathbb{R}$. By this, we give a full solution of the open problem formulated by Menshikov and Volkov [Urn-related random walk with drift $\rho x^{\alpha }\mathrm{/}{t}^{\beta }$’ \emph{Electron. J. Probab.}, $\mathbf{\text{13}}$ (2008), paper No. 31, 944–960] that was partially solved in the aforementioned paper by Abramov.