Suppose that we have a probability distribution $D$. Define $F_n$ to be the cumulative probability distribution function of the average of $n$ independent random variables drawn from $D$, and define $F_* (x) = \lim_{n\to \infty}F_n (x)$. I've become interested in the question of what functions $F_*$ might possibly be. I've come up with two classes of functions and three individual functions that $F_*$ might be equal to, but I haven't yet been able to determine whether any others are possible.
If the Central Limit Theorem applies to $D$, and it has mean $\mu$ and standard deviation $\sigma$, then for sufficiently large values of $n$, $F_n$ is approximately the cumulative distribution function of $\mathcal{N}(\mu, \frac{\sigma}{\sqrt n})$. So in this case, $F_*(x)$ is $0$ when $x\lt\mu$, $1$ when $x\gt\mu$, and I'm pretty sure it's always $0.5$ when $x=\mu$.
If $D$ is a degenerate distribution which always takes the value $v$, then the average of any number of variables drawn from that distribution will always be $v$, so all the $F_n$ will be the same as the cumulative distribution function of the original distribution. Therefore, $F_*(x)$ is $0$ when $x\lt v$ and $1$ when $x\ge v$.
If $D$ is a distribution with infinite expected value, such as the payout of the game from the St. Petersburg paradox, then the average of $n$ values drawn from that distribution will become arbitrarily likely to exceed any specific number as $n\to \infty$, so $F_*(x)=0$ for all $x$.
Likewise, if $D$ has an expected value of negative infinity, then $F_*(x)=1$ for all $x$.
By modifying the game from the St. Petersburg paradox a bit, I think I can also get $F_*(x)=0.5$ for all $x$. (The basic idea is to make the payout grow so fast with the number of coin flips that the total of $n$ payouts tends to be approximately equal to the largest payout, then switch the payout to negative with 50% probability. I could go into detail about why I think this works, but it would probably double the size of this question.)
Other than these two classes of functions and three individual constant functions, I haven't been able to find any other likely candidates, but I also haven't been able to find any reason why it should be impossible for $F_*$ to be anything else, as long as $0\le F_*(x)\le 1$ for all $x$ and $F_*$ is monotonically nondecreasing. (I do at least suspect that $F_*(x)$ is always well-defined for all $x$, but I'm not sure how to prove that the values of $F_n(x)$ must converge for all $x$ no matter what distribution we choose, so I could be wrong.)
So my question is: Can we conclude anything else about what kinds of functions are or are not possible for $F_*$? In particular: Is it possible for $F_*(x)$ to be anything other than $0$, $0.5$, or $1$? Is it possible for $F_*(x)$ to be strictly greater than zero and less than one in a region containing more than one value of $x$, without having to expand that region to the whole number line?