Martingale Convergence and the Functional Equation in the Multi-Type Branching Random Walk

Kyprianou, Andreas E.; Sani, Alireza

doi:10.2307/3318727

Cited by 16 publications

(25 citation statements)

References 17 publications

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“…The one-type branching random walk has received extensive treatment in the literature. The multitype extension has received less attention, but discussion of it can be found in Mode (1971), Biggins (1976), (1996), Bramson et al (1992), and Kyprianou and Rahimzadeh Sani (2001).…”

Section: Introductionmentioning

confidence: 99%

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Convergence results on multitype, multivariate branching random walks

Biggins

Sani

2005

Adv. Appl. Probab.

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We consider a multitype branching random walk on d-dimensional Euclidian space. The uniform convergence, as n goes to infinity, of a scaled version of the Laplace transform of the point process given by the nth generation particles of each type is obtained. Similar results in the one-type case, where the transform gives a martingale, were obtained in Biggins (1992) and Barral (2001). This uniform convergence of transforms is then used to obtain limit results for numbers in the underlying point processes. Supporting results, which are of interest in their own right, are obtained on (i) 'Perron-Frobenius theory' for matrices that are smooth functions of a variable λ ∈ L and are nonnegative when λ ∈ L − ⊂ L, where L is an open set in C d , and (ii) saddlepoint approximations of multivariate distributions. The saddlepoint approximations developed are strong enough to give a refined large deviation theorem of Chaganty and Sethuraman (1993) as a by-product.

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Section: Introductionmentioning

confidence: 99%

“…The mean convergence of {W n i (λ)} for real λ was also discussed in Bramson et al (1992), Kyprianou and Rahimzadeh Sani (2001), and, rather briefly, in Biggins and Kyprianou (2004 (λ), where W i (λ) is the limit of the martingale {W n i (λ)} as n → ∞. The sequences {W n i (λ)} and {W n ij (λ)} converge on the set 2 introduced in Theorem 5.…”

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confidence: 96%

Convergence results on multitype, multivariate branching random walks

Biggins

Sani

2005

Adv. Appl. Probab.

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“…More generally, similar changes of measure for other types of branching processes have become increasingly common in the study of classical and modern branching processes; in particular, the reader is referred to Lyons et al [29] and Lyons [30]. Champneys et al [7], Harris and Williams [19], Olofsson [34], Athreya [2], Kyprianou and Rahimzadeh Sani [28], Biggins and Kyprianou [3], Engländer and Kyprianou [13] and Kuhlbusch [26] also provide further examples of their use.…”

Section: Spine Decompositions For Bbmmentioning

confidence: 99%

Further probabilistic analysis of the Fisher–Kolmogorov–Petrovskii–Piscounov equation: one sided travelling-waves

Harris

Kyprianou

2006

Annales de l'Institut Henri Poincare (B) Probability and Statis

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At the heart of this article will be the study of a branching Brownian motion (BBM) with killing, where individual particles move as Brownian motions with drift −ρ, perform dyadic branching at rate β and are killed on hitting the origin.Firstly, by considering properties of the right-most particle and the extinction probability, we will provide a probabilistic proof of the classical result that the 'one-sided' FKPP travelling-wave equation of speed −ρ with solutions f : [0, ∞) → [0, 1] satisfying f (0) = 1 and f (∞) = 0 has a unique solution with a particular asymptotic when ρ < √ 2β, and no solutions otherwise. Our analysis is in the spirit of the standard BBM studies of Harris [18] and Kyprianou [27] and includes an intuitive application of a change of measure inducing a spine decomposition that, as a by product, gives the new result that the asymptotic speed of the right-most particle in the killed BBM is √ 2β − ρ on the survival set. Secondly, we introduce and discuss the convergence of an additive martingale for the killed BBM, W λ , that appears of fundamental importance as well as facilitating some new results on the almost-sure exponential growth rate of the number of particles of speed λ ∈ (0, √ 2β − ρ). Finally, we prove a new result for the asymptotic behaviour of the probability of finding the right-most particle with speed λ > √ 2β − ρ. This result combined with Chauvin and Rouault's [9] arguments for standard BBM readily yields an analogous Yaglom-type conditional limit theorem for the killed BBM and reveals W λ as the limiting Radon-Nikodým derivative when conditioning the right-most particle to travel at speed λ into the distant future.

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“…Also see Lyons et al [17,13,18] and other recent work based on these (examples are Kyprianou [14], Kyprianou and Sani [15], Athreya [1], Olofsson [19] amongst others) for similar spine based approaches in branching processes.…”

Section: The Spine Approach Martingales and Measuresmentioning

confidence: 99%

“…Git et al [4] presented alternative heuristic arguments based on birth-death processes to arrive at the expression (15). Using Euler-Lagrange techniques they showed that the specific path…”

Section: Large Deviations Heuristicsmentioning

confidence: 99%

Some Path Large-Deviation Results for a Branching Diffusion

Hardy

Harris

2013

Springer Proceedings in Mathematics &Amp; Statistics

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We give an intuitive proof of a path large-deviations result for a typed branching diffusion as found in Git, J.Harris and S.C.Harris [4]. Our approach involves an application of a change of measure technique involving a distinguished infinite line of descent, or spine, and we follow the spine setup of Hardy and Harris [5,7,6]. Our proof combines simple martingale ideas with applications of Varadhan's lemma, and is successful mainly because a 'spine decomposition' effectively reduces otherwise extremely difficult calculations on the whole collection of branching-diffusion particles down to just a single diffusing particle (the spine) whose large-deviations behaviour is well known. A similar approach was first used for branching Brownian motion in Hardy and Harris [6]. Importantly, our techniques should be applicable in a much wider class of branching diffusion large-deviation problems.

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Martingale Convergence and the Functional Equation in the Multi-Type Branching Random Walk

Cited by 16 publications

References 17 publications

Convergence results on multitype, multivariate branching random walks

Convergence results on multitype, multivariate branching random walks

Further probabilistic analysis of the Fisher–Kolmogorov–Petrovskii–Piscounov equation: one sided travelling-waves

Some Path Large-Deviation Results for a Branching Diffusion

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