Branching Random Walks: École d'Été de Probabilités de by Zhan Shi

By Zhan Shi

Providing an hassle-free advent to branching random walks, the main target of those lecture notes is at the asymptotic homes of one-dimensional discrete-time supercritical branching random walks, and particularly, on severe positions in each one new release, in addition to the evolution of those positions over the years.

Starting with the straightforward case of Galton-Watson bushes, the textual content basically concentrates on exploiting, in a number of contexts, the spinal constitution of branching random walks. The notes finish with a few purposes to biased random walks on trees.

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2 in Sect. 1/ 2 R. s. limit of Wn . s. 20) Proof of the Biggins Martingale Convergence Theorem Let Q be the probability on F1 such that QjFn D Wn PjFn for all n 0. 20) fails. 3) of Sect. W1 / D 0. To prove our claim, let us distinguish two possibilities. 1/. d. 5 in Sect. 6), so by the law of large numbers, when n ! wn / ! s. 1/ ! s. 1/ ! s. 1/, the associated random walk . s. s. , as claimed. W1 lnC W1 / D 1. d. s. wn n / ! s. , as claimed. 1/ and EŒW1 lnC W1  < 1. wn / as well as the offspring of wn , for all n 0.

X//C . s. finite. 4)). u C ˛/C . x/ ˛. 2 Convergence of the Derivative Martingale 51 with c4 WD cc32 . 0; 1/ is a fixed real number. wk 1 / C ˛ : We now estimate the expectation term on the right-hand side. 4) of Sect. X; e X/. 1 C minfy; zg/ for some constant c5 > 0 and all y 0 and z 0. 5)). 8). 1 C lnC X/ < 1. 5. 2. 1/. x/  < 1. 3). Fix " > 0. Recall that P . / WD P. j non-extinction/. 1 of Sect. x/ ! s. x/ > ˛g 1 ". ˛/ gative martingale. s. Dn C ˛Wn / (when n ! 1). Since Wn ! 2 Convergence of the Derivative Martingale 53 obtain that with probability at least 1 ", Dn converges to a finite limit.

15 in Sect. ˛/ 3 Strictly speaking, we use its approximation Dn in order to get a positive measure. 4 In Biggins and Kyprianou [57], the set Z ŒA is called a “very simple optional line”. x/ 32 ln n converges weakly to a non-degenerate limiting distribution. As a warm up to the highly technical proof of this deep result, we devote this section to explaining why 32 ln n should be the correct centering term. Recall that P . / WD P. j non-extinction/. x/ ! ; ln n jxjDn 2 in probability, under P . The proof of the theorem relies on the following preliminary lemma, which is stated uniformly in z 2 Œ0; 32 ln n for an application in Sect.

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