I am wondering if anyone can hint the results in fCLT in Banach's continuous function space in a compact set $ S $ endowed with the maximum norm.

My problem is as follows: given $ X_1, …, X_N sim X $ iid processes I want to prove that the process

$$ frac {1} { sqrt {N}} sum_ {n = 1} ^ N r_n (X_n – bar X) $$

weakly converges to the gaussian process with the covariance function given by $ { rm cov} (X (s) X (t)) $. here $ bar X $ is the sample average and $ r_1, …, r_N sim r $ iid processes with zero mean and unit variance.

Using the Jain-Marcus theorem of (1) this is simple under these types of Hölder conditions. However, all results require independence of the addends.

Is there a way to avoid that? Asymptotically my addends are independent.

I'd be happy with suggestions or articles that address similar questions.

Thank you!

(1) Chang, C. and Ogden, R. T. (2009). Bootstrapping sums of independent but not identically distributed continuous processes with applications to functional data. Multivariate Analysis Journal, 100 (6), 1291-1303.