Using the framework of random walks in random scenery, Cohen and Samorodnitsky (2006) introduced a family of symmetric $\alpha$-stable motions called local time fractional stable motions. When $\alpha=2$, these processes are precisely fractional Brownian motions with $1/2 < H < 1$. Motivated by random walks in alternating scenery, we find a complementary family of symmetric $\alpha$-stable motions which we call indicator fractional stable motions. These processes are complementary to local time fractional stable motions in that when $\alpha=2$, one gets fractional Brownian motions with $0 < H < 1/2$.
"Indicator fractional stable motions." Electron. Commun. Probab. 16 165 - 173, 2011. https://doi.org/10.1214/ECP.v16-1611