I'm not sure what you're asking, but there are central limit theorems for sums of independent rvs.
There are all kinds of CLTs, when the rvs are not indep and even when the variance is infinite, which is surprising.
is there a way to approximate
Sigma (Yn)
where Yn is my random variable following geometric distribution with parameter pn
(n is large)
If p for all n were same then it could be approximated by negative binomial. Does anyone have idea how to solve this or how to get a bound for this problem if it can not be solved approximately