Kolmogorov's three-series theorem
Encyclopedia
In probability theory
, Kolmogorov's three-series theorem, named after Andrey Kolmogorov
, gives a criterion for the almost sure convergence
of an infinite series of random variable
s in terms of the convergence of three different series involving properties of their probability distribution
s.
be a sequence of independent random variables. Then the random series 
converges almost surely if and only if the following three conditions hold:
(Here,
denotes the expectation of a random variable 
; 
denotes the variance of 
;
and
denotes the indicator of an event 
). If one of the conditions does not hold, then 
diverges almost surely.
Here, "
" means that each term 
is taken with a random sign that is either 
or 
with respective probabilities 
, and all random signs are chosen independently. Letting 
in the theorem denote a random variable that takes the values 
and 
with equal probabilities, one can check easily that the conditions of the theorem are satisfied, so it follows that the harmonic series with random signs converges almost surely. On the other hand, the analogous series of (for example) square root reciprocals with random signs, namely
diverges almost surely, since condition (3) in the theorem is not satisfied. Note that this is different from the behavior of the analogous series with alternating signs,
, which does converge.
, can be used to give a relatively easy proof of the Strong Law of Large Numbers.
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
, Kolmogorov's three-series theorem, named after Andrey Kolmogorov
Andrey Kolmogorov
Andrey Nikolaevich Kolmogorov was a Soviet mathematician, preeminent in the 20th century, who advanced various scientific fields, among them probability theory, topology, intuitionistic logic, turbulence, classical mechanics and computational complexity.-Early life:Kolmogorov was born at Tambov...
, gives a criterion for the almost sure convergence
Convergence of random variables
In probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to statistics and stochastic processes...
of an infinite series of random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
s in terms of the convergence of three different series involving properties of their probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
s.
Statement of the theorem
Let be a sequence of independent random variables. Then the random series converges almost surely if and only if the following three conditions hold:
- The series
converges. 
(Here,
denotes the expectation of a random variable ; denotes the variance of ;and
denotes the indicator of an event ). If one of the conditions does not hold, then diverges almost surely.Example
As an illustration of the theorem, consider the example of the harmonic series with random signs:
Here, "
" means that each term is taken with a random sign that is either or with respective probabilities , and all random signs are chosen independently. Letting in the theorem denote a random variable that takes the values and with equal probabilities, one can check easily that the conditions of the theorem are satisfied, so it follows that the harmonic series with random signs converges almost surely. On the other hand, the analogous series of (for example) square root reciprocals with random signs, namely
diverges almost surely, since condition (3) in the theorem is not satisfied. Note that this is different from the behavior of the analogous series with alternating signs,
, which does converge.Other applications
Kolmogorov's three-series theorem, combined with Kronecker's lemmaKronecker's lemma
In mathematics, Kronecker's lemma is a result about the relationship between convergence of infinite sums and convergence of sequences. The lemma is often used in the proofs of theorems concerning sums of independent random variables such as the strong Law of large numbers...
, can be used to give a relatively easy proof of the Strong Law of Large Numbers.