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.
Let $n\ge2$, and let $X=(X_1,\ldots,X_n)$ be a centered Gaussian vector with $\mathrm{Var}(X_i)=1$ for every $i$. Let $Z_1,\ldots,Z_n$ be independent standard Gaussians, and put $\overline{Z}=(Z_1+\cdots+Z_n)/n$. We prove $\mathbb{P}\{\max_i X_i\le t\}\ge\mathbb{P}\{\sqrt{n/(n-1)}\,\max_i(Z_i-\overline{Z})\le t\}$ for...
Let \(d\ge2\) be fixed and let \(G_n\) be finite abelian groups of odd orders \(N_n\to\infty\). We determine the centered diameter-\(d\) critical window for the standard random Cayley graph in which each nonzero group element is selected independently. Writing \(M_n=(N_n-1)/2\), we prove that the normalized first dista...
We establish a nonasymptotic Gaussian approximation bound over centered Euclidean balls for sums of independent, not necessarily identically distributed, random vectors in \(\mathbb R^p\). Let \(S_n\) denote the sum, let \(\Sigma=\operatorname{Var}(S_n)\), and let \(G\sim N(0,\Sigma)\). Define the effective rank and th...
A. Naumov, Sergey Samsonov, Vladimir Spokoiny et al.· 0 citations
We show that for any positive operator monotone function $f$ the inequality $\int_{[0,\infty)} f(t) \, \mathrm{d} \left\lVert E^{\Delta_{\varphi,\psi}}(t) \xi_\psi \right\rVert^2 + f'_\infty \varphi(1-s(\psi)) \geq \frac{f(1)}{2} \Bigl( \varphi(1) + \psi(1) - \lVert \varphi - \psi \rVert \Bigr)$ holds, where $\varphi,...
Let $X=(X_1,\ldots,X_n)$ have independent coordinates with mean zero, variance one, and $\|X_i\|_{\psi_2}\le K$, and let $H_d=(\mathbb R^n)^{\otimes_2 d}$. Let $L>0$ and let $f:H_d\to\mathbb R$ be convex and $L$-Lipschitz. We prove that, for $0\le t\le c_KLn^{d/2}$, \[ \textsf{P}\left\{ \left\lvert f(X^{\otimes d})-\te...
Let $w \colon (0,1) \to \mathbb{R}_{+}$ be a weight. We prove that for an arbitrary Schauder basis $\{e^{i \lambda_n t}\}_{n \in \mathbb{Z}}$ in $L^2(0,1)$ and an arbitrary lacunary defect set $A \subset \mathbb{Z}$ the system $\{w(t) r_n(t)\}_{n \in \mathbb{Z} \setminus A}$ is always complete and never minimal for any...
I. Aref'ev, A. V. Semenov· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
