The Hartman–Wintner–Strassen law of the iterated logarithm states that if
X 1,
X 2,… are independent identically distributed random variables and
S n =
X 1+
???+
X n , then
$\limsup_{n}S_{n}/\sqrt{2n\log \log n}=1\quad \text{a.s.},\qquad \liminf_{n}S_{n}/\sqrt{2n\log \log n}=-1\quad \text{a.s.}$
if and only if
EX 1 2 =1 and
EX 1=0. We extend this to the case where the
X n are no longer identically distributed, but rather their distributions come from a finite set of distributions.