In this paper, we derive a uniform stochastic bound of the operator norm (or equivalently,the largest singular value)of random matrices whose elements are indexed by parameters. As an application, we propose a new estimator that minimizes the(\beta) that may depend on a possibly infinite-dimensional parameter \beta\in }\Bbf\E  ||X(\beta)|| \leq CK }\left( + \gamma_2(}\Bbf\right)(\cdot), and \gamma_2(}\Bbf\Bbf operator norm of the matrix that consists ofthe moment functions. We show the consistency of the estimator .
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 }\Bbf, we obtain a uniform bound on its operator norm of the form\E \sup_{\beta \in \Bbf ||X(\beta)|| \leq CK }\left(\sqrt{\max(N,T) + \gamma_2(}\Bbf,d_\Bbf)\right), where C is an absolute constant, K controls the tail behavior of (the increments of) x_{it(\cdot), and \gamma_2(}\Bbf,d_\Bbf) is Talagrand's functional, a measure of multi-scale complexity of the metric space (\Bbf,d_\Bbf). 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 .
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In this paper, we derive a uniform stochastic bound of the operator norm (or equivalently,the largest singular value)of random matrices whose elements are indexed by parameters. As an application, we propose a new estimator that minimizes the
For an N \times T random matrix X(\beta) with weakly dependent uniformly sub-Gaussian entries x_{it
A
, we obtain a uniform bound on its operator norm of the form
A
\sup_{\beta \in \Bbf
A
\sqrt{\max(N,T)
A
,d_\Bbf)
A
, where C is an absolute constant, K controls the tail behavior of (the increments of) x_{it
A
,d_\Bbf) is Talagrand's functional, a measure of multi-scale complexity of the metric space (
A
,d_\Bbf). We illustrate how this result may be used for estimation that seeks to minimize the
R
the matrix that consists ofthe moment functions. We show the consistency of the estimator
moment conditions as well as for estimation of the maximal number of factors with functional data