  We consider the one-dimensional totally asymmetric simple exclusion process
with an arbitrary initial condition in a spatially periodic domain, and obtain
explicit formulas for the multi-point distributions in the space-time plane.
The formulas are given in terms of an integral involving a Fredholm
determinant. We then evaluate the large-time, large-period limit in the
relaxation time scale, which is the scale such that the system size starts to
affect the height fluctuations. The limit is obtained assuming certain
conditions on the initial condition, which we show that the step, flat, and
step-flat initial conditions satisfy. Hence, we obtain the limit theorem for
these three initial conditions in the periodic model, extending the previous
work on the step initial condition. We also consider uniform random and
uniform-step random initial conditions.
