Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Martingale convergence and the stopped branching random walk
Download PDF
Download PDF
  • Published: March 2000

Martingale convergence and the stopped branching random walk

  • A.E. Kyprianou1 

Probability Theory and Related Fields volume 116, pages 405–419 (2000)Cite this article

  • 205 Accesses

  • 15 Citations

  • Metrics details

Abstract.

We discuss the construction of stopping lines in the branching random walk and thus the existence of a class of supermartingales indexed by sequences of stopping lines. Applying a method of Lyons (1997) and Lyons, Pemantle and Peres (1995) concerning size biased branching trees, we establish a relationship between stopping lines and certain stopping times. Consequently we develop conditions under which these supermartingales are also martingales. Further we prove a generalization of Biggins' Martingale Convergence Theorem, Biggins (1977a) within this context.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, The University of Edinburgh, James Clarke Maxwell Building, Kings Buildings, Mayfield Road, Edinburgh EH9 3JZ, UK. e-mail: andreas@maths.ed.ac.uk, , , , , , GB

    A.E. Kyprianou

Authors
  1. A.E. Kyprianou
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Received: 21 December 1998 / Revised version: 28 June 1999 / Published online: 7 February 2000

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Kyprianou, A. Martingale convergence and the stopped branching random walk. Probab Theory Relat Fields 116, 405–419 (2000). https://doi.org/10.1007/s004400050256

Download citation

  • Issue Date: March 2000

  • DOI: https://doi.org/10.1007/s004400050256

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Peris
  • Random Walk
  • Convergence Theorem
  • Martingale Convergence Theorem
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature