Three-series theorem

Kolmogorov three-series theorem, three-series criterion

For each $ s> 0 $, let $ \tau _ {s} $ be the truncation function $ \tau _ {s} ( x)= s $ for $ x \geq s $, $ \tau _ {s} ( x) = x $ for $ | x | \leq s $, $ \tau _ {s} ( x)= - s $ for $ x \leq - s $.

Let $ X _ {1} , X _ {2} \dots $ be independent random variables with distributions $ F _ {1} , F _ {2} ,\dots $. Consider the sums $ S _ {n} = X _ {1} + \dots + X _ {n} $, with distributions $ F _ {1} \star \dots \star F _ {n} $. In order that these convolutions $ F _ {1} \star \dots \star F _ {n} $ tend to a proper limit distribution $ F $ as $ n \rightarrow \infty $, it is necessary and sufficient that for all $ s> 0 $,

$$ \tag{a1 } \sum _ { k } {\mathsf P} \{ | X _ {k} | > s \} < \infty , $$

$$ \tag{a2 } \sum \mathop{\rm Var} ( X _ {k} ^ { \prime } ) < \infty , $$

$$ \tag{a3 } \sum _ { k= 1} ^ { n } {\mathsf E} ( X _ {k} ^ { \prime } ) \rightarrow m , $$

where $ X _ {k} ^ { \prime } = \tau _ {s} ( X _ {k} ) $.

This can be reformulated as the Kolmogorov three-series theorem: The series $ \sum X _ {k} $ converges with probability $ 1 $ 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