  Limit theorems for the time average of some observation functions in an
infinite measure dynamical system are studied. It is known that intermittent
phenomena, such as the Rayleigh-Benard convection and Belousov-Zhabotinsky
reaction, are described by infinite measure dynamical systems.We show that the
time average of the observation function which is not the $L^1(m)$ function,
whose average with respect to the invariant measure $m$ is finite, converges to
the generalized arcsine distribution. This result leads to the novel view that
the correlation function is intrinsically random and does not decay. Moreover,
it is also numerically shown that the time average of the observation function
converges to the stable distribution when the observation function has the
infinite mean.
