Bivariate Markov chains converging to Lamperti transform Markov Additive Processes

摘要

Motivated by various applications, we describe the scaling limits ofbivariate 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 aset of $\kappa$ types. The chain starts from an initial value $(n,i)\in \mathbbZ_+ \times \{1,\ldots,\kappa\}$, with $i$ fixed and $n \rightarrow \infty$, andtypically we will assume that the macroscopic jumps of the marginal $X$ arerare, i.e. arrive with a probability proportional to a negative power of thecurrent state. We also assume that $X$ is non-increasing. We then observedifferent asymptotic regimes according to whether the rate of type change isproportional to, faster than, or slower than the macroscopic jump rate. Inthese different situations, we obtain in the scaling limit Lamperti transformsof Markov additive processes, that sometimes reduce to standard positiveself-similar Markov processes. As first examples of applications, we study thenumber of collisions in coalescents in varying environment and the scalinglimits of Markov random walks with a barrier. This completes previous resultsobtained by Haas and Miermont as well as Bertoin and Kortchemski in themonotype setting. In a companion paper, we will use these results as a buildingblock to study the scaling limits of multi-type Markov branching trees, withapplications to growing models of random trees and multi-type Galton-Watsontrees.