  We show that for almost every map in a transversal one-parameter family of
piecewise expanding unimodal maps the Birkhoff sum of suitable observables
along the forward orbit of the turning point satisfies the law of iterated
logarithm. This result will follow from an almost sure invariance principle for
the Birkhoff sum, as a function on the parameter space. Furthermore, we obtain
a similar result for general one-parameter families of piecewise expanding maps
on the interval.
