  This paper gives various asymptotic formulae for the transition probability
associated with discrete time quantum walks on the real line. The formulae
depend heavily on the `normalized' position of the walk. When the position is
in the support of the weak-limit distribution obtained by Konno, one observes,
in addition to the limit distribution itself, an oscillating phenomenon in the
leading term of the asymptotic formula. When the position lies outside of the
support, one can establish an asymptotic formula of large deviation type. The
rate function, which expresses the exponential decay rate, is explicitly given.
Around the boundary of the support of the limit distribution (called the
`wall'), the asymptotic formula is described in terms of the Airy function.

Probability;Functional Analysis