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Symmetry in the Green’s Function for Birth-death Chains

Author

Listed:
  • Greg Markowsky

    (Monash University)

  • José Luis Palacios

    (The University of New Mexico)

Abstract

A symmetric relation in the probabilistic Green’s function for birth-death chains is explored. Two proofs are given, each of which makes use of the known symmetry of the Green’s functions in other contexts. The first uses as primary tool the local time of Brownian motion, while the second uses the reciprocity principle from electric network theory. We also show that the the second proof extends easily to cover birth-death chains (a.k.a. state-dependent random walks) on trees, and can be adapted in order to derive hitting times on trees.

Suggested Citation

  • Greg Markowsky & José Luis Palacios, 2019. "Symmetry in the Green’s Function for Birth-death Chains," Methodology and Computing in Applied Probability, Springer, vol. 21(3), pages 841-851, September.
    • Handle: RePEc:spr:metcap:v:21:y:2019:i:3:d:10.1007_s11009-017-9581-4
    DOI: 10.1007/s11009-017-9581-4
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    References listed on IDEAS

    as
    1. Markowsky, Greg, 2011. "Applying Brownian motion to the study of birth-death chains," Statistics & Probability Letters, Elsevier, vol. 81(8), pages 1173-1178, August.
    2. Palacios, JoséLuis & Tetali, Prasad, 1996. "A note on expected hitting times for birth and death chains," Statistics & Probability Letters, Elsevier, vol. 30(2), pages 119-125, October.
    3. José Luis Palacios & Daniel Quiroz, 2016. "Birth and Death Chains on Finite Trees: Computing their Stationary Distribution and Hitting Times," Methodology and Computing in Applied Probability, Springer, vol. 18(2), pages 487-498, June.
    Full references (including those not matched with items on IDEAS)

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