Some process, which could be random or deterministic, is defined as a sequence with the finite and discrete outcome $S(n)$ = {$X_i \in \Omega$ | $i = 1, 2, 3, \dotsi, n$}. For example, $\Omega = \{E_1, E_2, E_3\}$ that has 3 possible outcomes.
If the existence of the probability measure for each of the 3 outcomes is not known, would the frequency of each outcome in large numbers, defined as below, converge to a fixed probability? $$ f_i \equiv \lim\limits_{n \rightarrow \infty}\frac{N_i(n)}{n}, i = 1, 2, 3 $$
where $N_i(n)$ is the number of outcome $E_i$ $\in$ $S(n)$.
The law of large numbers does not apply here, because of at least two reasons.
- The existence of a fixed probability for each of the 3 outcomes is not given.
- The conditions for proving the law of large numbers are not guaranteed in the above process. For example, the independence among outcomes doesn't have to hold.
Is there any established mathematical theorem that can show that, under the more general conditions, there always exist a probability for each outcome, that the relative frequency in large numbers will converge to?