Bivariate Markov chains converging to Lamperti transform Markov Additive Processes

Motivated by various applications, we describe the scaling limits of bivariate Markov chains $(X,J)$ on $\mathbb Z_+ \times \{1,\ldots,\kappa\}$ where $X$ can be viewed as a position marginal and $\{1,\ldots,\kappa\}$ is a set of $\kappa$ types. The chain starts from an initial value $(n,i)\in \mathbb Z_+ \times \{1,\ldots,\kappa\}$, with $i$ fixed and $n \rightarrow \infty$, and typically we will assume that the macroscopic jumps of the marginal $X$ are rare, i.e. arrive with a probability proportional to a negative power of the current state. We also assume that $X$ is non-increasing. We then observe different asymptotic regimes according to whether the rate of type change is proportional to, faster than, or slower than the macroscopic jump rate. In these different situations, we obtain in the scaling limit Lamperti transforms of Markov additive processes, that sometimes reduce to standard positive self-similar Markov processes. As first examples of applications, we study the number of collisions in coalescents in varying environment and the scaling limits of Markov random walks with a barrier. This completes previous results obtained by Haas and Miermont as well as Bertoin and Kortchemski in the monotype setting. In a companion paper, we will use these results as a building block to study the scaling limits of multi-type Markov branching trees, with applications to growing models of random trees and multi-type Galton-Watson trees.