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We study a Markov chain on $\mathbb{R}_+ \times S$, where $\mathbb{R}_+$ isthe non-negative real numbers and $S$ is a finite set, in which when the$\mathbb{R}_+$-coordinate is large, the $S$-coordinate of the process isapproximately Markov with stationary distribution $\pi_i$ on $S$. If $\mu_i(x)$is the mean drift of the $\mathbb{R}_+$-coordinate of the process at $(x,i) \in\mathbb{R}_+ \times S$, we study the case where $\sum_{i} \pi_i \mu_i (x) \to0$, which is the critical regime for the recurrence-transience phasetransition. If $\mu_i(x) \to 0$ for all $i$, it is natural to study theLamperti case where $\mu_i(x) = O(1/x)$; in that case the recurrenceclassification is known, but we prove new results on existence andnon-existence of moments of return times. If $\mu_i (x) \to d_i$ for $d_i \neq0$ for at least some $i$, then it is natural to study the generalized Lamperticase where $\mu_i (x) = d_i + O (1/x)$. By exploiting a transformation whichmaps the generalized Lamperti case to the Lamperti case, we obtain a recurrenceclassification and existence of moments results for the former. The generalizedLamperti case is seen to be more subtle, as the recurrence classificationdepends on correlation terms between the two coordinates of the process.

Non-homogeneous random walks on a half strip with generalized Lamperti drifts