  We study multiplicative nested sums, which are generalizations of harmonic
sums, and provide a calculation through multiplication of index matrices.
Special cases interpret the index matrices as stochastic transition matrices of
random walks on a finite number of sites. Relations among multiplicative nested
sums, which are generalizations of relations between harmonic series and
multiple zeta functions, can be easily derived from identities of the index
matrices. Combinatorial identities and their generalizations can also be
derived from this computation.
