We derive Hanson-Wright inequalities for the quadratic form of a random vector with sparse independent components. Specifically, we consider cases where the components of the random vector are sparse $\alpha$-subexponential random variables with $\alpha>0$. Our proof relies on a novel combinatorial approach to estimate the moments of the random quadratic form. In addition, we obtain a new Bernstein-type inequality for the sum of independent sparse $\alpha$-subexponential random variables. We present two applications with the sparse Hanson-Wright inequality: (1) Local law and complete eigenvector delocalization for sparse $\alpha$-subexponential Hermitian random matrices; (2) Concentration of the Euclidean norm for the linear transformation of a sparse $\alpha$-subexponential random vector.

翻译：我们推导了具有稀疏独立分量的随机向量二次型的 Hanson-Wright 不等式。具体而言，我们考虑随机向量的分量是稀疏的 $\alpha$-次指数随机变量（其中 $\alpha>0$）的情形。我们的证明依赖于一种新颖的组合方法来估计随机二次型的矩。此外，我们为独立稀疏 $\alpha$-次指数随机变量之和获得了一个新的 Bernstein 型不等式。我们展示了稀疏 Hanson-Wright 不等式的两个应用：(1) 稀疏 $\alpha$-次指数 Hermitian 随机矩阵的局部定律和完全特征向量离域化；(2) 稀疏 $\alpha$-次指数随机向量线性变换的欧几里得范数集中性。