We provide a random walk in random scenery representation of a new class of stable self-similar processes with stationary increments introduced recently by Jung, Owada and Samorodnitsky. In the functional limit theorem they provided only a single instance of this class arose as a limit. We construct a model in which a significant portion of processes in this new class is obtained as a limit.
"Random walks in doubly random scenery." Electron. Commun. Probab. 23 1 - 11, 2018. https://doi.org/10.1214/18-ECP165