Does the existence of the mean of a bounded, real-valued sequence imply it has a limiting distribution? let's assume we have a sequence $a_k \in (c,d) \subset \mathbb{R}\;\forall k>0$  and that the mean of the sequence converges:
$$\lim_{n \to \infty}\frac{1}{n}\sum_{i=1}^n a_i = L$$
Does this imply that there exists a limiting distribution of the values of $a_k$?
Making this a bit more precise:
Let $s_k:= \sum_{i=1}^k a_i$ be the partial sums and $m_k := \frac{s_k}{k}$ be the k-th average.
Conjecture
$$ \lim_{k\to \infty} m_k < \infty \implies$$
$$ \forall I \subset (c,d) \;\;\lim_{k \to \infty} \frac{1}{k}\sum_{i=1}^k\mathbb{1}_{\in I}(a_i) = \mu(I) \in [0,1]$$
If this is true, it suggests the values of any bounded sequence that "converges in mean" implicitly defines a probability distribution, which (I hope) would allow one to leverage statistical and measure-theoretic arguments about this series.
So far I'm having a hard time constructing a counterexample.
Thanks!
 A: Choose $(x_n)$ and $(y_n)$ as bounded sequences for which
$$ \frac{1}{n}\sum_{k=1}^{n} \delta_{x_k} \xrightarrow[\text{as } n\to\infty]{d} \mu \qquad\text{and}\qquad \frac{1}{n}\sum_{k=1}^{n} \delta_{y_k} \xrightarrow[\text{as } n\to\infty]{d} \nu, $$
where $\mu$ and $\nu$ are probability measures on $\mathbb{R}$ having the same first moment. For example, we may choose $(x_n) = (0, 1, 0, 1, \ldots)$ and $(y_n) = (\frac{1}{2}, \frac{1}{2}, \ldots)$ so that $\mu = \frac{1}{2}(\delta_0 + \delta_1)$ and $\nu=\delta_{1/2}$.
Now let $(N_k)_{k\geq 1}$ be a sequence of positive integers such that $ N_{k+1}/N_k \to \infty$ as $k\to\infty$, and define
$$ (a_n)_{n\geq 1} = (x_1, \ldots, x_{N_1}) \oplus (y_1, \ldots, y_{N_2}) \oplus (x_{N_1+1}, \ldots, x_{N_3}) \oplus (y_{N_2+1}, \ldots, y_{N_4}) \oplus \cdots,  $$
where $\oplus$ is the concatenation operation. Then it is not hard to show that $\frac{1}{n}\sum_{k=1}^{n} a_k$ converges to the common first moment of $\mu$ and $\nu$, but $\frac{1}{n}\sum_{k=1}^{n} \delta_{a_k}$ does not converge in distribution.
