Generalized LIL for geometrically weighted random series in Banach spaces

Let be a sequence of independent and identically distributed random variables, taking values in a separable Banach space B with topological dual B⁎. Considering the geometrically weighted series for 0<β<1, motivated by Einmahl and Li (2005, 2008), a general law of the iterated logarithm for ξ(β) is established.