  In infinite ergodic theory, two distributional limit theorems are well-known.
One is characterized by the Mittag-Leffler distribution for time averages of
$L^1(m)$ functions, i.e., integrable functions with respect to an infinite
invariant measure. The other is characterized by the generalized arc-sine
distribution for time averages of non-$L^1(m)$ functions. Here, we provide
another distributional behavior of time averages of non-$L^1(m)$ functions in
one-dimensional intermittent maps where each has an indifferent fixed point and
an infinite invariant measure. Observation functions considered here are
non-$L^1(m)$ functions which vanish at the indifferent fixed point. We call
this class of observation functions weak non-$L^1(m)$ function. Our main result
represents a first step toward a third distributional limit theorem, i.e., a
distributional limit theorem for this class of observables, in infinite ergodic
theory. To prove our proposition, we propose a stochastic process induced by a
renewal process to mimic a Birkoff sum of a weak non-$L^1(m)$ function in the
one-dimensional intermittent maps.
