We answer in a probabilistic setting two questions raised by Stokolos in a private communication. Precisely, given a sequence of random variables $\left\{X_k : k \geq 1\right\}$ uniformly distributed in $(0,1)$ and independent, we consider the following random sets of directions\begin{equation*}\Omega_{\text{rand},\text{lin}} := \left\{ \frac{\pi X_k}{k}: k \geq 1\right\}\end{equation*}and\begin{equation*}\Omega_{\text{rand},\text{lac}} := \left\{\frac{\pi X_k}{2^k} : k\geq 1 \right\}.\end{equation*}

We prove that almost surely the directional maximal operators associated to those sets of directions are not bounded on $L^p({\mathbb{R}}^2)$ for any $1 \lt p \lt \infty$.

我们在概率设置中回答 Stokolos 在私人通信中提出的两个问题。准确地说，给定一系列随机变量 $\left\{X_k : k \geq 1\right\}$ 均匀分布在 < a i=3>$(0,1)$ 并且独立，我们考虑以下随机方向集\begin{equation*}\Omega_{\ text{rand},\text{lin}} := \left\{ \frac{\pi X_k}{k}: k \geq 1\right\}\end{方程*}和\begin{方程*}\Omega_{\text{rand},\text{lac}} := \left\{\frac{\pi X_k}{2^k} : k\geq 1 \right\}.\end{方程*}

我们证明几乎可以肯定与这些方向集相关的方向最大运算符不受 $L^p({\mathbb{R}}^2)$< 的限制/span>。$1 \lt p \lt \infty$ 对于任何