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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!

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  • $\begingroup$ 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? $\endgroup$
    – Dieter Kadelka
    Commented May 28, 2023 at 9:52
  • $\begingroup$ 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. $\endgroup$
    – happyle
    Commented May 28, 2023 at 9:59
  • $\begingroup$ 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. $\endgroup$
    – Thomas Kojar
    Commented May 29, 2023 at 3:36

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