Abstract: In this talk, we go over a few recent developments related to a positive and self-similar Markov process with explicit transition density function. We explain how the process arises as the tangent process at the boundary of a large family of Markov processes, including in particular the q-Brownian motions. We then show how the convergence to the tangent process plays a role in the limit fluctuations for height functions of random Motzkin paths and for cumulative density functions of open asymmetric simple exclusion processes in the steady state. It is known that the positive self-similar Markov process has also intrinsic connection to free probability; the talk however does not require any knowledge on free probability. The talk is based on joint works with Wlodek Bryc.