  We consider two classes of piecewise expanding maps $T$ of $[0,1]$: a class
of uniformly expanding maps for which the Perron-Frobenius operator has a
spectral gap in the space of bounded variation functions, and a class of
expanding maps with a neutral fixed point at zero. In both cases, we give a
large class of unbounded functions $f$ for which the partial sums of $f\circ
T^i$ satisfy an almost sure invariance principle. This class contains piecewise
monotonic functions (with a finite number of branches) such that: - For
uniformly expanding maps, they are square integrable with respect to the
absolutely continuous invariant probability measure. - For maps having a
neutral fixed point at zero, they satisfy an (optimal) tail condition with
respect to the absolutely continuous invariant probability measure.

Probability;Dynamical Systems