Let $f_i:R^m \to R, i = 1,...n$ be the real functions and $\{i_k\}$ be the sequence of random variables $i_k$, which is independently randomly chosen in $\{1,...,n\}$. Denote by $f(x) =\frac{1}{n}\sum_{i=1}^nf_i(x)$. Is there a constant $C$ such that $\mathbb{E}[\sup_x|\frac{1}{l}\sum_{k=1}^lf_{i_k}(x) - f(x)|] \leq \frac{C}{\sqrt{l}}$ ?
2026-02-22 21:51:46.1771797106
Uniform convergence of finite sum of functions
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