On the convergence of supercritical general (C-M-J) branching processes

Nerman, Olle

doi:10.1007/bf00534830

Cited by 216 publications

(387 citation statements)

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“…The following theorem is due to Nerman [22] and is an analogue of the renewal theorem for ordinary random walks.…”

Section: 2mentioning

confidence: 99%

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Lacunarity of self-similar and stochastically self-similar sets

Gatzouras

1999

Trans. Amer. Math. Soc.

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Abstract. Let K be a self-similar set in R d , of Hausdorff dimension D, and denote by |K( )| the d-dimensional Lebesgue measure of its -neighborhood. We study the limiting behavior of the quantity −(d−D) |K( )| as → 0. It turns out that this quantity does not have a limit in many interesting cases, including the usual ternary Cantor set and the Sierpinski carpet. We also study the above asymptotics for stochastically self-similar sets. The latter results then apply to zero-sets of stable bridges, which are stochastically self-similar (in the sense of the present paper), and then, more generally, to level-sets of stable processes. Specifically, it follows that, if Kt is the zero-set of a realvalued stable process of index α ∈ (1, 2], run up to time t, then −1/α |Kt( )| converges to a constant multiple of the local time at 0, simultaneously for all t ≥ 0, on a set of probability one.The asymptotics for deterministic sets are obtained via the renewal theorem. The renewal theorem also yields asymptotics for the mean E[|K( )|] in the random case, while the almost sure asymptotics in this case are obtained via an analogue of the renewal theorem for branching random walks.

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“…The following theorem is due to Nerman [22] and is an analogue of the renewal theorem for ordinary random walks.…”

Section: 2mentioning

confidence: 99%

“…Recall our standing assumption 1 < E[ν] < ∞. Nerman's theorem actually requires less than E[ν] < ∞; see [22] for the precise conditions needed. Furthermore, the lattice case does not require Z to have sample paths which possess right-and left-hand limits.…”

Section: Theorem 34 Suppose That There Exists a Non-increasing And mentioning

confidence: 99%

Lacunarity of self-similar and stochastically self-similar sets

Gatzouras

1999

Trans. Amer. Math. Soc.

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“…The branching process discussed in the previous section is still valid and would allow us to calculate the variance in the overall number of infected but the method is problematic to implement (Nerman, 1981;Ball and Donnelly, 1995). Instead, we investigate this phase of the epidemic by deriving a diffusion approximation for the process in the limit of the number of clumps m → ∞ The variance tends to decrease as we increase β past this threshold as large outbreaks become more probable.…”

Section: Early Asymptotic Behaviourmentioning

confidence: 99%

The effect of clumped population structure on the variability of spreading dynamics

Black

House

Keeling

et al. 2014

Journal of Theoretical Biology

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Processes that spread through local contact, including outbreaks of infectious diseases, are inherently noisy, and are frequently observed to be far noisier than predicted by standard stochastic models that assume homogeneous mixing. One way to reproduce the observed levels of noise is to introduce significant individual-level heterogeneity with respect to infection processes, such that some individuals are expected to generate more secondary cases than others. Here we consider a population where individuals can be naturally aggregated into clumps (subpopulations) with stronger interaction within clumps than between them. This clumped structure induces significant increases in the noisiness of a spreading process, such as the transmission of infection, despite complete homogeneity at the individual level. Given the ubiquity of such clumped aggregations (such as homes, schools and workplaces for humans or farms for livestock) we suggest this as a plausible explanation for noisiness of many epidemic time series.

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“…In the context of Crump-Mode-Jagers processes, φ can be interpreted as a general characteristic of the process, see [13]. As usual, we suppress the dependence of φ on ω in most of the formulas, i.e., we write φ(t) and think of it as the random variable ω → φ(t, ω …”

Section: As Renewal Theorems For Brws On the Linementioning

confidence: 99%

“…For instance, Nerman's a.s. renewal theorem [13,Theorem 5.4] has been successfully applied in the study of the functional equation in the BRW, see e.g. [4,Theorem 8.6] and [2,Proposition 9.2].…”

Section: Introductionmentioning

confidence: 99%