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A Note on the Quadratic Birth Process
Author(s) -
John P. W. M.
Publication year - 1961
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s1-36.1.159
Subject(s) - homogeneous , mathematics , quadratic equation , series (stratigraphy) , function (biology) , simple (philosophy) , process (computing) , combinatorics , computer science , geometry , philosophy , paleontology , epistemology , evolutionary biology , biology , operating system
It is well known that the time homogeneous birth process with parameters λ n is divergent in the sense that∑ n = 1 ∞p n ( t ) < 1 for all t > 0 if and only if the series ∑ λ n − 1converges (see, for example, Feller [ 2 ], John [ 3 ]). Thenp ∞ ( t ) = 1 − ∑ n = 1 ∞p n ( t )is the probability that an infinite number of events occur in finite time t . The quadratic birth process in which λ n = λ n 2 and n (0) = 1 is such a process. In this case p ∞ ( l ) may be obtained in a simple closed form as a Jacobi Theta function.