Dose McDiarmid's inequality holds in hilbert space,specially or in $L^2$ space? Consider McDiarmid's inequality in Hilbert space, or can we extend McDiarmid's inequality to functional data analysis with the complete observed data?

This depends a little on what you're going for. A few pointers:

*

*Right out of the box, McDiarmid gives you good pointwise estimates on the difference of functions $(\sum_{i} x_{i}(t) - E[X(t)])$ for fixed $t$. If you know something about the continuity of your functions, you can get something about the maximum error by bounding this at some collection of times $t_{1},\ldots,t_{k}$ and using a "net" argument.


*You may want a nice representation of $F_{X}$ in terms of driving i.i.d. random variables. This isn't going to be universal advice, but often "nice" functions $F_{X}$ can be decomposed in such a way that most of the variation occurs in early terms. If that works out, you can get a better "net" argument. McDiarmid does hold for Martingales: https://people.math.sc.edu/lu/papers/concen.pdf.


*If the net argument in (1) isn't good enough, you may want to use a maximal inequality. See e.g. https://en.wikipedia.org/wiki/Doob_martingale. Note that this looks like you get only $\frac{1}{n}$-style convergence like in Markov's inequality, but later in the article you see that you can get exponential convergence as well.
