# Can we use epsilon-net method on an open set?

Let $$X_{t\in T}$$ be a random function where $$T$$ is a subset of $$\mathbb{R}^n$$.

Since $$T$$ has inifite points, we are not able to use union bound to estimate $$\sup X_t$$. Thus instead, when $$T$$ is compact (then the covering number of $$T<+\infty$$), we could use $$\epsilon$$-net argument to derive a meaningful upper bound on $$P(\sup X_t\geq\cdots)\leq$$.

However in my case, $$T$$ is a rectangle with one open line, i.e. $$[0,1]^n\times (0,1]$$. I was wondering could we use $$\epsilon$$-net in this case?

Many thanks for suggestions!

• What is $\mathbb{P}(sup$ ... Please be more explicit. What do you know about $(X_t)_{t \in A}$? And what is an $\epsilon$-net argument? Commented May 28, 2023 at 9:52
• sorry for the confusion. $P(\sup f\geq \epsilon)$ is the probability that the supreme of a random function larger than the value $\epsilon$. $X_t$ is a general random process. $\epsilon$-net argument is a method to derive estimation on $\sup$ of random process over a set with infinite points. Commented May 28, 2023 at 9:59
• It will depend on the process because it could be that $X_{t}$ diverges at the boundary not included in $T$. If you have a specific process in mind, we can try to see if some modification of chaining or some other method will apply. Commented May 29, 2023 at 3:36