  By using the matrix formulation of the two-step approach to the distributions
of runs, a recursive relation and an explicit expression are derived for the
generating function of the joint distribution of rises and falls for
multivariate random sequences in terms of generating functions of individual
letters, from which the generating functions of the joint distribution of
rises, falls, and number of runs are obtained. An explicit formula for the
joint distribution of rises and falls with arbitrary specification is also
obtained.
