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Limit theorem for derivative martingale at criticality w.r.t branching Brownian motion

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  • Yang, Ting
  • Ren, Yan-Xia

Abstract

We consider a branching Brownian motion on in which one particle splits into 1+X children. There exists a critical value in the sense that is the lowest velocity such that a traveling wave solution to the corresponding Kolmogorov-Petrovskii-Piskunov equation exists. It is also known that the traveling wave solution with velocity is closely connected with the rescaled Laplace transform of the limit of the so-called derivative martingale . Thus special interest is put on the property of its limit . Kyprianou [Kyprianou, A.E., 2004. Traveling wave solutions to the K-P-P equation: alternatives to Simon Harris' probability analysis. Ann. Inst. H. Poincaré 40, 53-72.] proved that, if EX(log+X)2+[delta] 0 while if EX(log+X)2-[delta]=+[infinity]. It is conjectured that is non-degenerate if and only if EX(log+X)2

Suggested Citation

  • Yang, Ting & Ren, Yan-Xia, 2011. "Limit theorem for derivative martingale at criticality w.r.t branching Brownian motion," Statistics & Probability Letters, Elsevier, vol. 81(2), pages 195-200, February.
    • Handle: RePEc:eee:stapro:v:81:y:2011:i:2:p:195-200
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    Cited by:

    1. Hou, Haojie & Ren, Yan-Xia & Song, Renming, 2024. "1-stable fluctuation of the derivative martingale of branching random walk," Stochastic Processes and their Applications, Elsevier, vol. 172(C).

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