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On the unimodality of passage time densities in birth‐death processes
Author(s) -
Keilson J.
Publication year - 1981
Publication title -
statistica neerlandica
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.52
H-Index - 39
eISSN - 1467-9574
pISSN - 0039-0402
DOI - 10.1111/j.1467-9574.1981.tb00710.x
Subject(s) - unimodality , mathematics , ergodic theory , combinatorics , monotone polygon , convolution (computer science) , state (computer science) , birth–death process , plane (geometry) , pure mathematics , geometry , algorithm , population , demography , machine learning , sociology , artificial neural network , computer science
It has been shown [2] that for any ergodic birth‐death process the p.d.f. of T on , the passage time from the reflecting state 0 to any level n is log‐concave and hence strongly unimodal. It is also known (cf [2]) that the p.d.f. of T n, n+1 or T n+1, n for such a process is completely monotone and hence unimodal. It has been conjectured that the p.d.f. for the passage time T mn between any two states is unimodal. An analytical proof of the result is presented herein, based on underlying renewal structure and methods in the complex plane. It is further shown that the p.d.f. of T mn can always be written as the convolution of two p.d.f.s, one completely monotone and the second PF and hence log‐concave.