Generalized LIL for geometrically weighted random series in Banach spaces

Let { X , X n ; n ⩾ 0 } 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 ξ ( β ) = ∑ n = 0 ∞ β n X n for 0 < β < 1 , motivated by Einmahl and Li (2005, 2008), a general law of the iterated logarithm for ξ ( β ) is established.