Consider a sequence $(x_n)_{n\in\mathbb{N}}$ of 0s and 1s. The asymptotic frequency of 1s in $x$ is usually defined as:

$$f=\lim_{n\to+\infty} \frac{1}{n}{\sum_{i=0}^{n-1} x_i}$$

when this limit exists. But sometimes the limit does not exist yet it sounds "reasonable" to say that the asymptotic frequency still exists. Is there a general way to define asymptotic frequency so that it applies to a much broader class of sequences?.

For simplicity, we can focus only on defining $f=0$: how to define there are "infinitely more" 0s than 1s in the sequence. Maybe this could involve probability theory, maybe not. Of course the definition can't be invariant under any permutation since there is nothing to say in terms of cardinal except there are as many 1s as 0s: countably many.

The tricky example I have in mind is successive groups of length $2^k$. In each group the values are equal: all 0s or all 1s. There are infinitely many groups with value 1 but these become more and more sparse for example with frequency $1/k$. It looks like it:

1-00-1111-0000000- ....... groups of length $2^k$: most are 0, with sometimes a group of 1s.

If you compress visually each group into a single digit, it looks like:

101000010000000000000100000000000000000000000000000000000000000000001....

The limit does not exist as the average is greater than 1/2 infinitely many times. Yet, this example can be thought as the realization of a random sequence $(X_n)_{n\in\mathbb{N}}$ that tends to 0 in probability. Hence it could make sense saying the asymptotic frequency is 0.