New Conditions for Central Limit Theorems.

Percy A. Pierre

Expert InsightsPublished 1968

Simple, new, necessary and sufficient conditions for the convergence in distribution of sums of small independent random variables to a normal random variable. For the case of finite second moments, the most general known condition for such a central limit theorem is the Lindeberg condition; however, it is very difficult to apply analytically and even more difficult to test for validity in any given physical situation. This Paper presents new necessary and sufficient conditions for the case in which absolute moments of order greater than four are finite, conditions which involve only moments and are usually more tractable analytically. They can be tested on a given set of data by standard techniques of estimating moments. In addition to the conditions for normal convergence, moment conditions which imply convergence to a Poisson random variable are given. (See also P-3954.)

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Pierre, Percy A., New Conditions for Central Limit Theorems. Santa Monica, CA: RAND Corporation, 1968. https://www.rand.org/pubs/papers/P3953.html.
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