Consider McDiarmid's inequality in Hilbert space, or can we extend McDiarmid's inequality to functional data analysis with the complete observed data?
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$\begingroup$ Assume $n$ curves $x_1,x_2,\cdots,x_n$ are sampled from $F_X$, then we want to estimate $\sum_{i=1}^n x_i-E(X)$. $\endgroup$– ChenCommented Jun 21, 2022 at 7:24
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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.
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$\begingroup$ Thank you for your reply. I want to know the article you mentioned in 3. And in 1, if we can obtain the full observations but not the discrete estimated form $t_1,t_2,\cdots,t_k$. Dose McDiarmid also hold? $\endgroup$– ChenCommented Jun 22, 2022 at 12:36
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$\begingroup$ Sorry, I meant to refer to the same article as in 2:people.math.sc.edu/lu/papers/concen.pdf As for the other question: McDiarmid's inequality can't hold uniformly in t for "generic" functions $X(t)$, even if it holds pointwise for all individual $t$. You need some assumptions about the joint distribution. Continuity can work, as can others. $\endgroup$ Commented Jun 22, 2022 at 13:47