Bounded sequences with divergent Cesàro mean It is well known that there are bounded sequences with divergent Cesàro mean, i.e., a bounded $a_n$ for which given $$c_N :=  \frac{1}{N}\sum_{n=1}^N a_n,$$ the sequence $(c_N)_{N\geq1}$ has no limit. A simple example is $a_n = (-1)^{\lfloor \log_2 n\rfloor}$, for which the $(2^{n+1}-1)$-th term of the sequence of means is $-1/3$ for odd $n$, and the $(2^{n+1}-2)$-th is $1/3$ for even $n$.
But I don't know of any example which is substantially different.
My question is, then, can the Cesàro mean of a bounded sequence diverge in some other way or this slow but stubborn oscillation is essentially the only example?

EDIT:
Is there a simple characterization of the bounded sequences for which the Cesàro means converges?
 A: Choose any bounded infinite sequence $\{ b_m \}_{m \geq 0}$ of integers that is not eventually stationary, and let $a_n = b_{\lfloor \log (\log (n+2)) \rfloor}$.  When $m$ is a large integer, and $N+2$ is almost $e^{e^m}$, the Cesàro mean $c_N$ will be very close to $b_{m-1}$.
If you want arbitrary iterated Cesàro means to diverge, you can replace the double log with a slower-growing function, like the inverse Ackerman function $\alpha$.
Regarding your question, it is not hard to show that if your sequence is bounded, that the Cesàro means $c_N$ can't move very quickly for large $N$.  In other words, you can avoid oscillation in the literal sense, but whatever you have will be slow.
A: Here is a construction: Choose  real numbers a < c < d < b. Construct a sequence 
a, b, a, a, ..., a, b, b, ..., b, a, a, ..., a, b, b, ..., b, a, ......
consisting of blocks of a's and b's as follows: Start with the first two terms a, b. For the first block of a's put enough terms equal to a so that the Cesaro average so far is less that c. Then start inserting terms equal to b, enough of them so that the average so far exceeds d. Then start inserting terms equal to a until the average is less than c, and continue with inserting terms equal to b until the average exceeds d, and so on.  
