For an $N \times T$ random matrix $X(\beta )$ with weakly dependent uniformly sub-Gaussian entries $x_{it}(\beta )$ that may depend on a possibly infinite-dimensional parameter $\beta \in \mathbf {B}$ , we obtain a uniform bound on its operator norm of the form $\mathbb {E} \sup _{\beta \in \mathbf {B}} ||X(\beta )|| \leq CK \left (\sqrt {\max (N,T)} + \gamma _2(\mathbf {B},d_{\mathbf {B}})\right )$ , where C is an absolute constant, K controls the tail behavior of (the increments of) $x_{it}(\cdot )$ , and $\gamma _2(\mathbf {B},d_{\mathbf {B}})$ is Talagrand’s functional, a measure of multiscale complexity of the metric space $(\mathbf {B},d_{\mathbf {B}})$ . We illustrate how this result may be used for estimation that seeks to minimize the operator norm of moment conditions as well as for estimation of the maximal number of factors with functional data.

对于$N \times T$随机矩阵 $X(\beta )$ 具有弱依赖一致亚高斯条目 $x_{it}(\beta )$ 可能依赖于可能无限维参数 $\beta \in \mathbf {B}$ ，我们在其运算符范数上获得一个统一边界，形式 为 $\mathbb {E} \sup _{\beta \in \mathbf {B}} ||X(\beta )|| \leq CK \left (\sqrt {\max (N,T)} + \gamma _2(\mathbf {B},d_{\mathbf {B}})\right )$ ，其中C是绝对常数，K控制（增量） $x_{it}(\cdot)$ 和 $\gamma _2(\mathbf {B},d_{\mathbf {B}})$ 的尾部行为 是 Talagrand 的泛函，度量空间 $(\mathbf {B},d_{\mathbf {B}})$ 的多尺度复杂性的量度。我们说明了如何将此结果用于寻求最小化矩条件的算子范数的估计，以及如何使用函数数据估计最大因子数。