# Sum of random variables without central limit theorem

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.

• Direct convolution? Oct 7, 2015 at 16:14
• But in my case total number of variables are very large, like 1000. Oct 7, 2015 at 16:15
• What is the reason the Gaussian approximation is inadequate?
– guy
Oct 7, 2015 at 17:06
• @guy That's a great question. Note, however, that the different rates of the distributions require careful understanding and application of the CLT. If those rates vary a lot, then the CLT is likely to fail. Your suggestion, then, can be turned into actionable advice only through an analysis of how much the variances of those distributions vary, in order to determine whether a Gaussian approximation will meet whatever the OP's accuracy needs may be.
– whuber
Oct 7, 2015 at 17:14
• @whuber Right, I was probing for more information, not making any suggestion. If the concern is that there are a few whose rates are small enough to dominate the sum, then that is one thing. On the other hand, if the issue is that the data isn't iid because the rates vary (but are otherwise well behaved), then OP would presumably be interested in learning about the Lindeberg-Feller CLT.
– guy
Oct 7, 2015 at 17:18