In K. Yano, Y. Yano and M. Yor (2009), limit theorems for the one-dimensional symmetric $\alpha$-stable process normalized by negative (killing) Feynman-Kac functionals were studied. We consider the same problem and extend their results to positive Feynman-Kac functionals of multi-dimensional symmetric $\alpha$-stable processes.
"Feynman-Kac Penalisations of Symmetric Stable Processes." Electron. Commun. Probab. 15 32 - 43, 2010. https://doi.org/10.1214/ECP.v15-1524