Chapter

Stochastic Methods in Biology

Volume 70 of the series Lecture Notes in Biomathematics pp 210-221

Unimodality and Bounds of Modes for Distributions of Generalized Sojourn Times

  • Ken-iti SatoAffiliated withDepartment of Mathematics, College of General Education, Nogoya University

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Summary

The distribution of the generalized sojourn time T for a birth-and-death process up to the first passage time to the state n, starting at the state m (m < n), is considered. The generalized sojourn time is the sum of the sojourn times at the states i, over i = 0,1,...,n−1, with a state-dependent signed weight. A main new result of the paper is that the distribution of T is unimodal. Explicit description of the distribution is given by using exponential distributions on the positive and the negative axis. Bounds of the mode are derived from this description. Other bounds of the modes of general unimodal distributions are given in terms of absolute moments and central absolute moments. Infinite divisibility of T is also proved. These results are extended to generalized sojourn times of diffusion processes, which arise in models of neurobiology and population genetics.