Yes, that's what I meant. Thanks for editing my question.

@AnthonyQuas: Yes, the balls are in the Euclidean norm. I edited the question to clarify this. Thanks for your comment.

@IosifPinelis: Yes.

@CarloBeenakker: I finally went through your answer. I have a few questions: (i) since I assume that the Wigner matrix is normalized, why do you need to introduce the $n^{1/2}$ scaling in the first part of the proof? (ii) in which sense the summation converges to the integral in the large $n$ limit?

@CarloBeenakker: Okay, thanks. I edited my question to clarify notation.

Interesting approach, thanks. I'm missing something though: (i) in my question I assume that $i$ is fixed, e.g. $i=1$, and only $j$ varies, (ii) in your (double) sum to (double) integral transformation, isn't a factor $n$ missing?