Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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 are 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 $$ \lim_{n\to\infty} \frac{1}{n} \sum_{i=1}^n b_i = \lim_{n\to\infty} \frac{1}{n} \sum_{i=1}^n b_{\pi(i)}$$

share|improve this question
    
Do you really want the set S of permutations preserving a given Cesaro mean (thus S depends on the sequence $(b_i)\\ $), or the set of permutations that perserve any Cesaro mean ? –  Pietro Majer Jan 29 '13 at 22:11
    
I do mean the set $S$ of permutations preserving a given Cesaro mean. –  Daniel Mansfield Jan 31 '13 at 4:44
1  
Permutations preserving Cesaro mean for any sequence would be the Levy group. See theorem 2 of M. Blümlinger; N. Obata "Permutations preserving Cesàro mean, densities of natural numbers and uniform distribution of sequences" (1991) –  Daniel Mansfield Jan 31 '13 at 11:13

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.