I know that using central limit theorem we approximate sum of random variables into Gaussian distribution. Is the any other approximation method available for finding the probability distribution function of sum of independent random variables ? Note that the total number of variables are very large and the random variables has exponential distribution with different rates. Please let me know.
There is the Edgeworth series, which can be thought of as an adjustment to the central limit theorem that uses the cumulants of your variables.
Somewhat related to the answer by Hong Oi, there is saddlepoint approximations, which can be used in most cases where the central limit theorem can be used, and then gives much more accurate approximations. Saddlepoint methods are somewhat better behaved than Edgeworth series; they can never give a negative density, as is possible with Edgeworth. For an example, see my answer to General sum of Gamma distributions