by J.D.Biggins and N.H.Bingham
Math. Proc. Camb. Phil. Soc. (1991), 110, 545-558.
The occurrence of certain `near-constancy phenomena' in some aspects of the theory of (simple) branching processes forms the background for the work below. The problem arises out of work by Karlin and McGregor (1968a,b). A detailed study of the theoretical and numerical aspects of the Karlin-McGregor near-constancy phenomenon was given by Dubuc (1982), and considered further by Bingham (1988). We give a new approach which simplifies and generalises the results of these authors. The primary motivation for doing this was the recent work of Barlow and Perkins (1988), who observed near-constancy in a framework not immediately covered by the results then known.
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