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' Bull. Aust. Math. Soc. 109 (2024), 393–402] for the case of time-inhomogeneous random walks, the increments of which take values in ℝ. By this, we give a full solution of the open problem formulated by Menshikov and Volkov [`Urn-related random walk with drift ρ x^α/t^β' Electron. J. Probab., 13 (2008), paper No. 31, 944–960] that was partially solved in the aforementioned paper by Abramov.