In the physics literature, these are known as random walks with shrinking steps. The distribution for $\alpha = {\frac{1}{2}}$, the golden ratio, and $\alpha = 2 ^ {\frac{1}{n}}$, where n is a positive integer and the $X_i$'s are +1 or -1, are discussed in Krapivsky and Redner's American Journal of Physics article. Some possibly useful references are:

Serino, C.A. and Redner, S.,"The Pearson walk with shrinking steps in two dimensions",Journal of Statistical Mechanics: Theory and Experiment, volume = 1,01006 (2010)

Rador, T.,"Random walkers with shrinking steps in d dimensions and their long term memory",Physical Review E,74,051105,(2006)

Rador, T. and Taneri, S.,"Random walks with shrinking steps: First-passage characteristics", Physical Review E 73, 036118 (2006)

Krapivsky, P.L. and Redner, S.,"Random walk with shrinking steps",American Journal of Physics 72,591-598 (2004)