I am looking for a concentration inequality of a double sum...

Let $X_1,\dots, X_n$ be iid r.v. and also let $Y_1,\dots ,Y_n$ be iid such that even $X_i$ and $Y_j$ are idependent.

I am looking for a high probabylity bound for the event

$$ A=\left\{\left\|\frac{1}{n^2}\sum_{i=1}^n\sum_{j=1}^n f(X_i,Y_j)-E_{X,Y}(f(X,Y))\right\|_H\leq...\right\}\,, $$

where f is some bounded function in a Hilberspace H.

I am looking for a concentration inequality of a double sum….

Let $X_1,\dots, X_n$ be iid r.v. and also let $Y_1,\dots ,Y_n$ be iid such that even $X_i$ and $Y_j$ are independent.

I am looking for a high probability bound for the event

$$ A=\left\{\left\|\frac{1}{n^2}\sum_{i=1}^n\sum_{j=1}^n f(X_i,Y_j)-E_{X,Y}(f(X,Y))\right\|_H\leq\cdots\right\}\,, $$

where $f$ is some bounded function in a Hilbert space $H$.