  About two dozens of exactly solvable Markov chains on one-dimensional finite
and semi-infinite integer lattices are constructed in terms of convolutions of
orthogonality measures of the Krawtchouk, Hahn, Meixner, Charlier, $q$-Hahn,
$q$-Meixner and little $q$-Jacobi polynomials. By construction, the stationary
probability distributions, the complete sets of eigenvalues and eigenvectors
are provided by the polynomials and the orthogonality measures. An interesting
property possessed by these stationary probability distributions, called
`convolutional self-similarity,' is demonstrated.
