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Given a summable sequence $a_i$ of real numbers, theorems of Levi and later Agnew characterize the permutations $\pi: \mathbb N \mapsto \mathbb N$ which preserve this limitare sum preserveing: that is

$$\lim_{n\to\infty}\sum_{i=1}^n a_{i} = \lim_{n\to\infty}\sum_{i=1}^n a_{\pi(i)}$$

I would like to know of any similar research into permutations that preserve the Cesaro mean. That is, given a sequence $b_i \in \ell^\infty$, is there any characterization of permutations which $$\lim_{n\to\infty} \frac{1}{n} \sum_{i=1}^n b_i = \lim_{n\to\infty} \frac{1}{n} \sum_{i=1}^n b_{\pi(i)}$$

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Permutations that preserve Cesaro mean

Given a summable sequence $a_i$ of real numbers, theorems of Levi and later Agnew characterize the permutations $\pi: \mathbb N \mapsto \mathbb N$ which preserve this limit: that is

$$\lim_{n\to\infty}\sum_{i=1}^n a_{i} = \lim_{n\to\infty}\sum_{i=1}^n a_{\pi(i)}$$

I would like to know of any similar research into permutations that preserve the Cesaro mean. That is, given a sequence $b_i \in \ell^\infty$, is there any characterization of permutations which $$\lim_{n\to\infty} \frac{1}{n} \sum_{i=1}^n b_i = \lim_{n\to\infty} \frac{1}{n} \sum_{i=1}^n b_{\pi(i)}$$