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