D. Kendall. Biometrika, (1948)The author discusses a ``birth-and-death process'' for which the Kolmogorov differential equations assume the form $$ P_n'(t)=-n(a+b)+cP_n(t)+(n-1)a+cP_n-1(t)\\ +(n+1)bP_n+1(t), $$ where $P_n(t)$ is the probability of a population size $n$. The case $c0$ corresponds to mortality and fertility proportional to the actual population size. The $c$-term accounts for an increase by immigration. The generating function of $P_n(t)$ is obtained and it is shown that for small $c$ one obtains approximations to R. A. Fisher's ``logarithmic series distribution'' which has found several applications in biology..
D. Blackwell, и D. Kendall. J. Appl. Probability, (1964)Consider an ``urn scheme'' in which balls of $k$ colors are present in a single urn (initially one of each color) and successive random drawings made. After each drawing, the selected ball is replaced together with another of the same color. The authors add to the existing supply of examples by determining the Martin boundary of this process, which turns out to be homeomorphic to the set of $k$-vectors with non-negative components summing to 1. Applications to a moment problem and a stochastic birth process are discussed. Reviewer's remark: In case $k=2$, the boundary is the unit interval, as in T. Watanabe's coin-tossing example J. Math. Soc. Japan 12 (1960), 192--206; MR0120683 (22 #11432). In fact, the Pólya process is an $h$-process for coin-tossing, so that this agreement is no coincidence..