• Title/Summary/Keyword: branching Brownian motion

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LAW OF LARGE NUMBERS FOR BRANCHING BROWNIAN MOTION

  • Kang, Hye-Jeong
    • Journal of the Korean Mathematical Society
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    • v.36 no.1
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    • pp.139-157
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    • 1999
  • Consider a supercritical Bellman-Harris process evolving from one particle. We superimpose on this process the additional structure of movement. A particle whose parent was at x at its time of birth moves until it dies according to a given Markov process X starting at x. The motions of different particles are assumed independent. In this paper we show that when the movement process X is standard Brownian the proportion of particles with position $\leq${{{{ SQRT { t} }}}} b and age$\leq$a tends with probability 1 to A(a)$\Phi$(b) where A(.) and $\Phi$(.) are the stable age distribution and standard normal distribution, respectively. We also extend this result to the case when the movement process is a Levy process.

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