J. D. Biggins (1977). Martingale convergence in the branching random walk. J. Appl. Probability 14, no. 1, 25--37. Home/Search Document Not in Database Summary Related Articles Check |
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Branching Processes - Athreya, Vidyashankar (1999) (113 citations) (Correct)
....fW n ( n 1g is a non negative martingale sequence and hence converges w.p. 1 to a random variable W ( Furthermore, for 2 E(0) P (W ( 0jZ (0) ffi 0 ) 1 or q according as E(W 1 ( log W 1 ( jZ (0) ffi 0 ) 1 or 1: 2 The details of proof can be found in Biggins (see [28 ]) where the assumption m( 1 for all is considerably reduced. A conceptual proof of the above result along the lines of the one in [49] has been established by Lyons (see [50] Biggins [30] also considers uniform convergence of W n ( to W ( for in some compact set. Biggins and ....
Biggins,J.D.(1977): Martingale convergence in the branching random walk, Journal of Applied Probability, No. 1, 25-37.
The Smoothing Transform; the Boundary Case - Biggins, Kyprianou (2001) Self-citation (Biggins) (Correct)
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Biggins, J.D. (1977). Martingale convergence in the branching random walk. J. Appl. Probab. 14, 25-37. 9
A Simple Path to Biggins' Martingale Convergence for Branching.. - Lyons (1995) (2 citations) Self-citation (Biggins) (Correct)
....e GammaffX i # 1991 Mathematics Subject Classification. Primary 60J80. Key words and phrases. Galton Watson. Research partially supported by the Institute for Mathematics and Its Applications (Minneapolis) and NSF Grant DMS 9306954. 1 when this exists in [ Gamma1; 1] as a Lebesgue integral. Biggins (1977) has determined when W (ff) is nontrivial: Biggins Theorem. Suppose that ff 2 R is such that m(ff) 1 and m 0 (ff) exists and is finite. Then the following are equivalent: i) P[W (ff) 0] q; ii) P[W (ff) 0] 1; iii) E[W (ff) 1; iv) E[hff; Li log hff; Li] 1 and ffm 0 ....
.... 1 and m 0 (ff) exists and is finite. Then the following are equivalent: i) P[W (ff) 0] q; ii) P[W (ff) 0] 1; iii) E[W (ff) 1; iv) E[hff; Li log hff; Li] 1 and ffm 0 (ff) m(ff) log m(ff) Remark. In fact, the hypotheses here are very slightly weaker than those of Biggins (1977), Lemma 5. Moreover, the proof to follow works without the assumption that m 0 (ff) be finite, except for the implication (ii) iv) where it needs the assumption that ffm 0 (ff) 6= Gamma1. Remark. The case of Biggins Theorem where L is constant, X i are independent and identically ....
Biggins, J. D. (1977). Martingale convergence in the branching random walk. J. Appl. Prob. 14, 25--37.
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Biggins, J.D. (1977a). Martingale convergence in the branching random walk. Jnl. Appl. Probab. 14, 25--37.
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Biggins, J.D. (1977a). Martingale convergence in the branching random walk. J. Appl. Probab. 14, 25--37.
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Biggins, J.D. (1977). Martingale convergence in the branching random walk. J. Appl. Prob. 14, 25-37.
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Biggins, J.D. (1977a). Martingale convergence in the branching random walk. J. Appl. Probab. 14, 25-37.
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J. D. Biggins (1977). Martingale convergence in the branching random walk. J. Appl. Probability 14, no. 1, 25--37.
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John Biggins. Martingale convergence in the branching random walk. J. Appl. Probability, 14(1):25-37, 1977.
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Biggins, J.D. (1977) Martingale convergence in the branching random walk. J. Appl. Probab. 14, 25-37.
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Biggins, J.D. (1977) Martingale convergence in the branching random walk. J. Appl. Probab. 14, 25-37.
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Biggins, J.D. (1977) Martingale convergence in the branching random walk. J. Appl. Probab. 14, 25-37.
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