  We consider a two-state quantum walk on a line where after the first step an
absorbing sink is placed at the origin. The probability of finding the walker
at position $j$, conditioned on that it has not returned to the origin, is
investigated in the asymptotic limit. We prove a limit theorem for the
conditional probability distribution and show that it is given by the Konno's
density function modified by a pre-factor ensuring that the distribution
vanishes at the origin. In addition, we discuss the relation to the problem of
recurrence of a quantum walk and determine the Polya number. Our approach is
based on path counting and stationary phase approximation.
