If two sequences $\{a_k\}$ and $\{b_k\}$ are such that
$\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}(a_k-b_k)=0$
$\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}(a_k^2-b_k^2)=0$
does it mean that the two sequences are asymptotically equally distributed? i.e., you can't distinguish one sequence from the other?
EDIT
Assume that two sequences are such that
$\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}(a_k^q-b_k^q)=0;\ \forall\ q\in\mathbb{N}$
I'm asking this because a sentence in a paper claims that for two asymptotically equally distributed sequences, the above holds. I'm wondering if the converse is true, i.e., if the above hold for any sequence, are they asymptotically equally distributed?
EDIT 2
Definition: Asymptotically equally distributed sets
Two sets $\{a_k\}$ and $\{b_k\}$ are said to be asymptotically equally distributed if
$-\infty<\alpha_1\leq \{a_k,b_k\}\leq\alpha_2<\infty$
and
$\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}\left[f(a_k)-f(b_k)\right]=0$
where $f(x)$ is any function continuous on $[\alpha_1,\alpha_2]$. This is called Weyl's theorem or Weyl's definition, but I don't remember the original paper this was discussed.