In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
, Kolmogorov's Three-Series Theorem, named after
Andrey Kolmogorov
Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...
, gives a criterion for the
almost sure
In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. ...
convergence
Convergence may refer to:
Arts and media Literature
*''Convergence'' (book series), edited by Ruth Nanda Anshen
* "Convergence" (comics), two separate story lines published by DC Comics:
**A four-part crossover storyline that united the four Wei ...
of an
infinite series
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
of
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
s in terms of the convergence of three different series involving properties of their
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
s. Kolmogorov's three-series theorem, combined with
Kronecker's lemma In mathematics, Kronecker's lemma (see, e.g., ) 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 s ...
, can be used to give a relatively easy proof of the
Strong Law of Large Numbers
In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials shou ...
.
Statement of the theorem
Let
be
independent random variables
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independ ...
. The random series
converges almost surely
In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. ...
in
if the following conditions hold for some
, and only if the following conditions hold for any
:
Proof
Sufficiency of conditions ("if")
Condition (i) and
Borel–Cantelli give that
for
large,
almost surely
In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. ...
. Hence
converges if and only if
converges. Conditions (ii)-(iii) and
Kolmogorov's Two-Series Theorem give the almost sure convergence of
.
Necessity of conditions ("only if")
Suppose that
converges almost surely.
Without condition (i), by Borel–Cantelli there would exist some
such that
for infinitely many
, almost surely. But then the series would diverge. Therefore, we must have condition (i).
We see that condition (iii) implies condition (ii):
Kolmogorov's two-series theorem along with condition (i) applied to the case
gives the convergence of
. So given the convergence of
, we have