I want to generate an infinite sequence of numbers between $0$ and $9$ such that the percentage of number $i$ appearing in the sequence is $p_i$. Let $p=\lbrace p_0,...,p_9\rbrace$.
Another agent $B$ will observe $k$ consecutive elements from the sequence. Then it will update its belief about the probability of $p_i$ using a belief update rule (let's say a simple frequentist, i.e., the number of occurrences of each digit divided by $k$).
The question is how can I generate the sequence such that $B$'s belief $q$ is close to $p$, i.e., $d(p,q)$ is minimized where $d(p,q)$ is the dsitance function of two distributions, e.g., $d(p,q)=(\sum_i (p_i-q_i)^2)^0.5$.
Note that I don't know how $B$ will choose the $k$ numbers. Let's consider the worst case (i.e., worst $p'$).