I am seeking references for a particular setting of concentration inequality. I am deliberately being vague on some assumptions (especially on the input spaces), as I am interested in any setting.
Let $(X_i)_{i=1}^n$ be i.i.d. random variables distributed as a given random variable $X$. For all $i \in \{1,...,n \}$, let $(Y_{ip})_{p=1}^m$ be i.i.d random variables distributed as a given random variable $Y$. Let $f$ be a function with values in a separable Hilbert space $\mathcal H$. Supposing that the $X$s and $Y$s variables are independent as well, has anyone come across some kind of concentration inequality to control with high probability a quantity of the form: $$ \left \| \frac 1 n \sum_{i=1}^n \left ( \frac 1 m \sum_{p=1}^m f(X_i, Y_{ip}) - \mathbb E[f(X, Y)|X=X_i] \right ) \right \|_{\mathcal H} $$
Many thanks,
I have found a way which works for my application so I post it here in case it may interest someone:
Note that the random variables $(W_p)_{p=1}^m$ with $W_p := \frac 1 n \sum_{i=1}^n f(X_i, Y_{ip}) - \mathbb E[f(X, Y)|X=X_i]$ are independent conditional on the draws of the random variables $(X_i)_{i=1}^n$ and centered with respect to the expectation conditional on those draws as well
Then, provided we have a way to control the moments of the random variables $(W_p)_{p=1}^m$, we can apply the Hilbert-valued exponential concentration inequality given by Theorem 3.3.4 in Sums and Gaussian Vectors, Yurinsky [1995] to the empirical mean $\frac 1 p \sum_{p=1}^m W_p$ with respect to the probability conditional on the draws of the $(X_i)_{i=1}^n$. Then, the $(X_i)_{i=1}^m$ can be integrated out in the obtained inequality on probabilities.