Lower Deviation Probabilities for Level Sets of the Branching Random Walk

Shuxiong Zhang ()
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Shuxiong Zhang: Southern University of Science and Technology

Journal of Theoretical Probability, 2023, vol. 36, issue 2, 811-844

Abstract: Abstract Given a supercritical branching random walk $$\{Z_n\}_{n\ge 0}$$ { Z n } n ≥ 0 on $${\mathbb {R}}$$ R , let $$Z_n(A)$$ Z n ( A ) be the number of particles located in a set $$A\subset {\mathbb {R}}$$ A ⊂ R at generation n. It is known from Biggins (J Appl Probab 14:630–636, 1977) that under some mild conditions, for $$\theta \in [0,1)$$ θ ∈ [ 0 , 1 ) , $$n^{-1}\log Z_n([\theta x^* n,\infty ))$$ n - 1 log Z n ( [ θ x ∗ n , ∞ ) ) converges almost surely to $$\log \left( {\mathbb {E}}[Z_1({\mathbb {R}})]\right) -I(\theta x^*)$$ log E [ Z 1 ( R ) ] - I ( θ x ∗ ) as $$n\rightarrow \infty $$ n → ∞ , where $$x^*$$ x ∗ is the speed of the maximal position of $$\{Z_n\}_{n\ge 0}$$ { Z n } n ≥ 0 and $$I(\cdot )$$ I ( · ) is the large deviation rate function of the underlying random walk. In this work, we investigate its lower deviation probabilities, in other words, the convergence rates of $$\begin{aligned} {\mathbb {P}}\left( Z_n([\theta x^* n,\infty ))

Keywords: Branching random walk; Level sets; Lower deviation; 60F10; 60J80; 60G50 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10959-022-01183-z

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