I want to use generic chaining/ Dudley's integral to bind the below stochastic process \mathbb E\sup_{tin T}\frac{1}{n}\sum_{i=1}^nX_{i,t} where X_{i,t} takes value either 1 or 0 (binary random variable; T is a finite set). I want to use ||X_{i,t}-X_{i,t'}||_{\infty} ( the sup norm of the difference) as my distance metric. Is it possible to do generic chaining using this distance metric given the binary behaviour of the random variables?
I know that in chaining the discretization is on the set T. But my issue is as I go on refining the discretizations as in even if t and t' grow closer ( or as in walk along the chain to 't' via finer approximations) ||X_{i,t}-X_{i,t'}||_{\infty} this difference doesn't decrease ( it keeps on oscillating between 0 and 1).
Can I still do chaining in this scenario?
Any help will be highly appreciated.
