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In probability theory, Kolmogorov's two-series theorem is a result about the convergence of random series. It follows from Kolmogorov's inequality and is used in one proof of the strong law of large numbers.
Statement of the theorem
[edit]
Let
be independent random variables with expected values
and variances
, such that
converges in
and
converges in
. Then
converges in
almost surely.
Assume WLOG
. Set
, and we will see that
with probability 1.
For every
,
Thus, for every
and
,
While the second inequality is due to Kolmogorov's inequality.
By the assumption that
converges, it follows that the last term tends to 0 when
, for every arbitrary
.
- Durrett, Rick. Probability: Theory and Examples. Duxbury advanced series, Third Edition, Thomson Brooks/Cole, 2005, Section 1.8, pp. 60–69.
- M. Loève, Probability theory, Princeton Univ. Press (1963) pp. Sect. 16.3
- W. Feller, An introduction to probability theory and its applications, 2, Wiley (1971) pp. Sect. IX.9