A large-deviation principle for the empirical distribution of a regular branching random walk

Abstract

Let \(\{Z_n\}_{n\geq0}\) be a supercritical branching random walk with deterministic rooted \(b\)-ary tree, \(b\geq2\), and symmetric displacements satisfying \(\lim_{x\to+\infty}x^{-α}\log\Pp(X>x)=-λ\) with \(α,λ>0\). Set ${\overline{Z}}_n(\cdot ):=Z_n(\cdot )/Z_n(\mathbb{R}),n\ge 0.$ For a finite union \(A\) of intervals, we establish the following full large-deviation principle: for every Borel set \(Γ\subset[0,1]\), \begin{align*} -\inf_{q\inΓ^\circ}Q_A(q) &\leq \liminf_{n\to\infty}n^{-α/2} \log\Pp\!\left(\overline Z_n(\sqrt n\,A)\inΓ\right) &\leq \limsup_{n\to\infty}n^{-α/2} \log\Pp\!\left(\overline Z_n(\sqrt n\,A)\inΓ\right) &\leq -\inf_{q\in\overlineΓ}Q_A(q), \end{align*} where $Q_A$ is a good rate function on $[0,1]$. Meanwhile, we obtain large deviation probabilities for $\{{\overline{Z}}_n(\sqrt{n}A){\}}_{n\ge 1}.$ This strengthens the nonmatching upper and lower bounds obtained by Chen and He [Probab. Theory Related Fields 175 (2019) 255-307] for regular trees with Weibull displacements. Our method combines a large-deviation principle for rescaled displacement tree fields and exponential equivalence.

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