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

RESUMO

Let \(\{Z_n\}_{n\geq0}\) be a supercritical branching random walk with deterministic rooted \(b\)-ary tree, \(b\geq2\), and symmetric displacements satisfying \(\lim_{x\to+\infty}x^{-\alpha}\log\Pp(X>x)=-\lambda \) with \(\alpha,\lambda>0\). Set $\overline Z_n(\cdot):=Z_n(\cdot)/Z_n(\mathbb{R}),~n\geq0.$ For a finite union \(A\) of intervals, we establish the following full large-deviation principle: for every Borel set \(\Gamma\subset[0,1]\), \begin{align*} -\inf_{q\in\Gamma^\circ}Q_A(q) &\leq \liminf_{n\to\infty}n^{-\alpha/2} \log\Pp\!\left(\overline Z_n(\sqrt n\,A)\in\Gamma\right) &\leq \limsup_{n\to\infty}n^{-\alpha/2} \log\Pp\!\left(\overline Z_n(\sqrt n\,A)\in\Gamma\right) &\leq -\inf_{q\in\overline\Gamma}Q_A(q), \end{align*} where $Q_A$ is a good rate function on $[0,1]$. Meanwhile, we obtain large deviation probabilities for $\{\overline Z_n(\sqrt n\,A)\}_{n\geq1}.$ 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.