Before revision:  First, employing a carefully crafted coupling approach , we develop uniform distributional approximations for  t-statistic processes (indexed over the support of the conditioning variables). These results cover both undersmoothed and bias corrected inference, require seemingly minimal rate restrictions , and achieve fast approximation rates . Using the uniform approximations we construct valid confidence bands which share these same advantages.
After revision:  First, employing a carefully crafted coupling approach , we develop uniform distributional approximations for t-statistic processes and construct valid confidence bands. In the univariate case, our uniform distributional approximations require seemingly minimal rate restrictions , and achieve fast approximation rates . Using the uniform approximations we construct valid confidence bands which share these same advantages.