Abstract
For birth-and-death processes, we show that every initial distribution is reproduced from the first hitting time distribution. The reproduction is done by applying to the distribution function a differential operator defined through the eigenfunction of the generator. Using the spectral theory for generalized second-order differential operators, we study asymmetric random walks and binary branching processes.
Funding Statement
The research of Kosuke Yamato was supported by JSPS KAKENHI Grant Numbers JP21J11000, JP23J00817 and by JSPS Open Partnership Joint Research Projects grant no. JPJSBP120209921. The research of Kouji Yano was supported by JSPS KAKENHI grant no.’s JP19H01791 and JP19K21834 and by JSPS Open Partnership Joint Research Projects grant no. JPJSBP120209921.
Acknowledgements
The authors would like to thank Professor Patrick Fitzsimmons for drawing their attention to Rogers (1984). This research was supported by RIMS and by ISM.
Citation
Kosuke Yamato. Kouji Yano. "Reproduction of initial distributions from the first hitting time distribution for birth-and-death processes." Bernoulli 30 (2) 936 - 960, May 2024. https://doi.org/10.3150/23-BEJ1619
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