Averaging i.i.d. variables: Equal chance to be right and left of the mean? Let $\{X_i\}_{i=1}^{\infty}$ be i.i.d. random variables. Define 
$$ L_n = \frac{1}{n}\sum_{i=1}^n X_i \quad \forall n \in \{1, 2, 3, …\} $$
Using the central limit theorem, it can be shown that if $E[X_i]=0$ and $0<Var(X_i)<\infty$ then: 
$$ \lim_{n\rightarrow\infty} P[L_n\leq x] =  \left\{ \begin{array}{ll}
1 &\mbox{ if $x > 0$} \\
c & \mbox{ if $x=0$}\\
0 & \mbox{ if $x<0$} 
\end{array}
\right.$$
where $c=1/2$.  If the variance is infinite then the law of large numbers implies a similar structure for the cases $x>0$ and $x<0$, but the case $x=0$ is unclear to me. 
Questions: For infinite variance, can we get different behavior for the case $x=0$, such as $c=1/3$?  Can we get related step-function structure when the mean does not exist, but with different behavior for the case $x=0$?  

Notes:


*

*We can get such a limiting function with $c=1/3$ for random sequences with different structure, such as $L_n= A/n$ with $P[A=1]=2/3, P[A=-1]=1/3$.

*I came up with this question while reflecting on the question here: Why does a C.D.F need to be right-continuous?

Partial answer: if $X_i$'s are non-negative with infinite  mean than $L_n \to \infty$ a.s. an $P(L_n \leq x) \to 0$ for every $x$.
