ON THE FREQUENCY COUNT FOR A RANDOM-WALK WITH ABSORBING BOUNDARIES - A CARCINOGENESIS EXAMPLE .1.

A non-homogeneous random walk on non-negative integers with transition probabilities p0i = delta0i, p(Ni) = delta(Ni), P(i,i+1) = lambda(i), p(i,i-1) = mu(i), and p(i,i) = rho(i), lambda(i) + mu(i) + rho(i) = 1, is studied. In particular, when the transition probabilities are independent of position, a general expression for the joint probability generating function (JPGF) of the frequency count of the stages 1,2,...N - 1 is derived. The appropriate marginal forms of this JPGF yield the PGF of the frequency count at any pair of stages, and at any particular single stage. Some moment formulae associated with the frequency count are derived. A random walk conditional on absorption at a specified boundary is also considered. The random walk model proposed is eminently suitable for the example of carcinogenesis.