  We analyze the probability distributions of the quantum walks induced from
Markov chains by Szegedy (2004). The first part of this paper is devoted to the
quantum walks induced from finite state Markov chains. It is shown that the
probability distribution on the states of the underlying Markov chain is always
convergent in the Cesaro sense. In particular, we deduce that the limiting
distribution is uniform if the transition matrix is symmetric. In the cases of
non-symmetric Markov chain, we exemplify that the limiting distribution of the
quantum walk is not necessarily identical with the stationary distribution of
the underlying irreducible Markov chain. The Szegedy scheme can be extended to
infinite state Markov chains (random walks). In the second part, we formulate
the quantum walk induced from a lazy random walk on the line. We then obtain
the weak limit of the quantum walk. It is noted that the current quantum walk
appears to spread faster than its counterpart-quantum walk on the line driven
by the Grover coin discussed in literature. The paper closes with an outlook on
possible future directions.
