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Three-series theorem

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Kolmogorov three-series theorem, three-series criterion

For each , let be the truncation function for , for , for .

Let be independent random variables with distributions . Consider the sums , with distributions . In order that these convolutions tend to a proper limit distribution as , it is necessary and sufficient that for all ,

(a1)
(a2)
(a3)

where .

This can be reformulated as the Kolmogorov three-series theorem: The series converges with probability if (a1)–(a3) hold, and it converges with probability zero otherwise.

References

[a1] M. Loève, "Probability theory" , Princeton Univ. Press (1963) pp. Sect. 16.3
[a2] W. Feller, "An introduction to probability theory and its applications" , 2 , Wiley (1971) pp. Sect. IX.9
How to Cite This Entry:
Three-series theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Three-series_theorem&oldid=25947