limiting distribution of discrete pdf Assume that   x1,x2.....xn are i.i.d random variables with $$Xi = \left\{ \matrix{
   - 1\,,\,\,\,\,\,with\,probability\,\,\,of{1 \over 3}\hfill \cr 
  1\,\,\,\,\,\,\,\,\,with\,\,\,\,probability\,\,\,of {2 \over 3} \hfill \cr}  \right\}$$
determine the value of c for which $${{{2^{\sum\limits_{i = 1}^n {{X_i}} \,}}} \over n} \to \,\,c$$ in probability
my try
I know using W.L.L.N that $${{\sum\limits_{i = 1}^n {{X_i}} } \over n}\buildrel P \over
 \longrightarrow E(X)$$
so i tried to find the pdf of (2^x) as follows
$$\eqalign{
  & P({2^X} \le x)\, = \,P(X\ln 2 \le \ln x)\,  \cr 
  & P(X \le {{\ln x} \over {\ln 2}})\,\, = \,F({{\ln x} \over {\ln 2}}) \cr} $$
but i faced a problem when find pdf which is the derivative of the CDF
the pdf ive got is as follows
$$\left\{ \matrix{
  {2 \over {3\ln 2}}\,\,\,\,\,\,\,x = 0.5 \hfill \cr 
   \hfill \cr 
  {2 \over {6\ln 2}}\,\,\,\,\,\,\,x = 2 \hfill \cr}  \right\}$$
$$i\,know\,i\,did\,something\,wrong\,because\,the\,sum\,of\,the\,probalities\,in\,the\,pdf \ne \,1$$
Any help please??????

If $X = \sum^n_i X_i$, then it can take on any values in $\left\{-n,-\left(n-1\right),...,n-1,n\right\}$. It has a pmf, not a pdf, and the form of it would be much more complicated than shown here.
More importantly, are you sure there is a finite limit? If it exists, then
$\sum^n_{i=1} X_i - \log_2 n \rightarrow \log_2 c.$
But, from the strong law of large numbers, $\sum^n_{i=1} X_i$ grows linearly a.s.
