Recall the KMT embedding for empirical processes:
For all $n\geq 1$, there exists a probability space with $(U_k)_{1\leq k\leq n}$ i.i.d with uniform $[0, 1]$ distribution and $W_0$ a Brownian bridge such that for all $x>0$,
$\sup_{0\leq t\leq 1} |\frac{1}{\sqrt{n}} \sum_{k=1}^n (1_{U _k ≤t}-t) − W_0 (t)| ≤ C\frac{\log n + x}{\sqrt{n}}$ with probability at least $1 − \mathrm{e}^{-x}$, where $C$ is a universal constant.
Does there exist a version with the partial sums, i.e. for $1\leq m\leq n$, $\frac{1}{\sqrt{n}} \sum_{k=1}^m (1_{U _k ≤t}-t)$? More precisely, is there some process $W_1:[0,1]^2 \rightarrow \mathbb{R}$ such that $\sup_{\substack{0\leq t\leq 1 \\ 1\leq m \leq n}} |\frac{1}{\sqrt{n}} \sum_{k=1}^m (1_{U _k ≤t}-t) − W_1 (t,\frac{m}{n})| ≤ C'\frac{\log n + x}{\sqrt{n}}$ with probability at least $1 − \mathrm{e}^{-x}$?
That would be helpful in some Longest Increasing Subsequences problems, for example