I have a math/stat problem where I need to show the convergence of the average of a sequence of experiments to an interval. The sequence of experiments is not i.i.d., hence the standard law of large number does not apply. However, the framework satisfies some assumptions which might facilitate the convergence proof. I think the question can fit this advanced forum because it seems to go beyond standard applications of probability results.

Suppose we have a  sequence of random experiments $(a_n)_{n\in \mathbb{N}}$. In particular, each $a_n$ is a random draw from a probability distribution $P_n: B\rightarrow [0,1]$, where $B$ is a finite set. 

The  probability distributions $P_n$ are potentially different across $n$. However, for each $b\in B$ and $n\in \mathbb{N}$, we know that $P_n(b)\in [\nu_\ell(b), \nu_u(b)]$, where the latter interval does not vary across $n$.

Let $x_N(b):=\frac{1}{N}\sum_{n=1}^N \mathbb{1}(a_n=b)$ for a finite $N\in \mathbb{N}$, where $\mathbb{1}(a_n=b)$ takes value 1 if $a_n=b$ and 0 otherwise.  

I would like to show that, as $N\rightarrow \infty$, $x_N(b)$ falls in $[\nu_\ell(b), \nu_u(b)]$. 

Could you help me to do that? If you think the statement is wrong, can you explain why?

**Note: the draws may not be independent.**