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Law of the iterated logarithm for oscillating random walks ..:

Hambly, B.M. ; Kersting, G. ; Kyprianou, A.E.
Stochastic Processes and their Applications.  108 (2003)  2 - p. 327-343 , 2003
Link: https://doi.org/10.1016/j.spa.2003.07.002