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One-dimensional stepping stone models, sardine genetics and Brownian local time

Publication ,  Journal Article
Durrett, R; Restrepo, M
Published in: Annals of Applied Probability
February 1, 2008

Consider a one-dimensional stepping stone model with colonies of size M and per-generation migration probability v, or a voter model on ℤ in which interactions occur over a distance of order K. Sample one individual at the origin and one at L. We show that if Mv/L and L/K2 converge to positive finite limits, then the genealogy of the sample converges to a pair of Brownian motions that coalesce after the local time of their difference exceeds an independent exponentially distributed random variable. The computation of the distribution of the coalescence time leads to a one-dimensional parabolic differential equation with an interesting boundary condition at 0. © Institute of Mathematical Statistics, 2008.

Duke Scholars

Published In

Annals of Applied Probability

DOI

EISSN

1050-5164

ISSN

1050-5164

Publication Date

February 1, 2008

Volume

18

Issue

1

Start / End Page

334 / 358

Related Subject Headings

  • Statistics & Probability
  • 4905 Statistics
  • 4901 Applied mathematics
  • 0104 Statistics
  • 0102 Applied Mathematics
 

Citation

APA
Chicago
ICMJE
MLA
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Durrett, R., & Restrepo, M. (2008). One-dimensional stepping stone models, sardine genetics and Brownian local time. Annals of Applied Probability, 18(1), 334–358. https://doi.org/10.1214/07-AAP451
Durrett, R., and M. Restrepo. “One-dimensional stepping stone models, sardine genetics and Brownian local time.” Annals of Applied Probability 18, no. 1 (February 1, 2008): 334–58. https://doi.org/10.1214/07-AAP451.
Durrett R, Restrepo M. One-dimensional stepping stone models, sardine genetics and Brownian local time. Annals of Applied Probability. 2008 Feb 1;18(1):334–58.
Durrett, R., and M. Restrepo. “One-dimensional stepping stone models, sardine genetics and Brownian local time.” Annals of Applied Probability, vol. 18, no. 1, Feb. 2008, pp. 334–58. Scopus, doi:10.1214/07-AAP451.
Durrett R, Restrepo M. One-dimensional stepping stone models, sardine genetics and Brownian local time. Annals of Applied Probability. 2008 Feb 1;18(1):334–358.

Published In

Annals of Applied Probability

DOI

EISSN

1050-5164

ISSN

1050-5164

Publication Date

February 1, 2008

Volume

18

Issue

1

Start / End Page

334 / 358

Related Subject Headings

  • Statistics & Probability
  • 4905 Statistics
  • 4901 Applied mathematics
  • 0104 Statistics
  • 0102 Applied Mathematics