Here is the question, I am quite new to the field of Asymptotic Statistics and I'm kind of confused by the different definitions of stochastic equicontinuity in various resources. In particular, the stochastic equicontinuity is defined as follows in the book of Stochastic Limit Theory, a sequence is said to be asymptotically uniformly stochastically equicontinuous if for all $\varepsilon >0, \eta>0, \exists \delta>0$ such that
$\limsup_{n \rightarrow \infty} P[\sup_{\theta \in \boldsymbol{\Theta}} \sup_{d(\theta^{'},\theta)<\delta} |\mathbb{G}_n(\theta^{'})-\mathbb{G}_n(\theta)| > \varepsilon] < \eta$
while in many other resources, including the Asymptitotic Statistics by A:W: van der Vaart, we have a slightly different definition,
$\limsup_{n \rightarrow \infty} P[ \sup_{d(\theta^{'},\theta)<\delta} |\mathbb{G}_n(\theta^{'})-\mathbb{G}_n(\theta)| > \varepsilon] < \eta$
where the term $\sup_{\theta \in \boldsymbol{\Theta}}$ is removed, I wonder if the two definitions are still equivalent ? Could you clarify what's the role of this term, $\sup_{\theta \in \boldsymbol{\Theta}}$?