Uniform Law of large numbers Could you please help with the proof of the proposition:
Let $G(\cdot)$ be bounded, continuous, strictly increasing function on $\mathbb{R}.$ Let ${\xi_t},\, t\in\mathbb{Z}_+$ be i.i.d random variables.
Would it be true to say that we have something like the uniform Law of large numbers here? Why?
$$
\sup_{x\in\mathbb{R}}\;\left|\;n^{-1}\sum_{t=1}^n G(x+\xi_t) - \mathbf{E}G(x+\xi_1)\;\right| \stackrel{P}\longrightarrow 0, \quad n\to\infty.
$$
Thanks in advance!
 A: Probably an application of Glivenko-Cantelli's theorem as I suggested in the comments will work. 
Assume that the $\xi_t$ are defined on a probability space $(\Omega,\mathcal F,\mu)$. We can assume that $G$ is the cumulative distribution function of a real valued random variable. Let $(\eta_t,t\in\Bbb Z)$ be a collection of independent random variables with cumulative distribution function $G$, defined on $(\Omega',\mathcal G,\nu)$. Using Glivenko-Cantelli's theorem for the product space and the family of i.i.d. random variables $(\xi_t-\eta_t,t\in\Bbb Z)$ gives 
$$\sup_{u\in\Bbb R}\left|\frac 1n\sum_{i=1}^n\chi_{\{\eta_i(\omega')-\xi_i(\omega)\leqslant u\}}-\mu\otimes\nu\{\eta_i(\omega')-\xi_i(\omega)\leqslant u\}\right|\to 0$$
 for $\mu\otimes\nu$ almost every $(\omega,\omega')$. Then integrate with respect to $\nu$.
An alternative approach would use the result of the paper 
Ramon van Handel, The universal Glivenko-Cantelli property, Probab. Th. Rel. Fields 155, 911-934 (2013)
which is available here. We consider $\mathcal F:=\{t\mapsto G(x+t), x\in\Bbb R)\}$, which is a Glivenko-Cantelli class as $G$ is bounded. But it seems quite overkill for this problem, since their result is more general.
