  The discrete-time version of totally asymmetric simple-exclusion process
(TASEP) on a finite one-dimensional lattice is studied with the periodic
boundary condition. Each particle at a site hops to the next site with
probability $0 \leq p \leq 1$, if the next site is empty. This condition can be
rephrased by the condition that the number $n$ of vacant sites between the
particle and the next particle is positive. Then the average velocity is given
by a product of the hopping probability $p$ and the probability that $n \geq
1$. By mapping the TASEP to another driven diffusive system called the
zero-range process, it is proved that the distribution function of vacant sites
in the stationary state is exactly given by a factorized form. We define
$k$-particle velocity correlation function as the expectation value of a
product of velocities of $k$ particles in the stationary distribution. It is
shown that it does not depend on positions of $k$ particles on a circle but
depends only on the number $k$. We give explicit expressions for all velocity
correlation functions using the Gauss hypergeometric functions. Covariance of
velocities of two particles is studied in detail and we show that velocities
become independent asymptotically in the thermodynamic limit.
