Expected Kolmogorov complexity under Kolmogorov complexity distribution If $K(w)$ is the Kolmogorov complexity of a string $w$, where programs are prefix-encoded so $\sum_{w} 2^{-K(w)} \leq 1$, what is

$$\lim_{n\to\infty} \frac{\sum_{|w|=n}2^{-K(w)} K(w)}{\sum_{|w|=n} 2^{-K(w)}}?$$

Also, what is

$$\lim_{n\to\infty} \frac{\sum_{|w|=n}2^{-K(w)} \frac{K(w)}{n}}{\sum_{|w|=n} 2^{-K(w)}}?$$



The distribution here is closely related to the "universal distribution".
The second limit would say whether $\Theta(|w|)$ is an approximation for $K(w)$.
If $\alpha$ is the answer to the 1st question then $\alpha=\infty$. Namely, for any  $c $  there is an $n $ such that all strings $w $ of length at least $n $ have $K (w) \ge c$. In particular the expectation of $K (w) $ with respect to any distribution on strings of length $n $ is  $\ge c $.

Similarly if $\beta$ is the answer to the 2nd question then $0\le\beta\le 1$, since
$$
(\exists c)(\forall w)(K(w)\le |w|+2\log |w| + c),
$$
but I don't know exactly what $\beta$ is.