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 ...
, Lindeberg's condition is a
sufficient condition
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...
(and under certain conditions also a necessary condition) for the
central limit theorem
In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themsel ...
(CLT) to hold for a sequence of independent
random variables
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 ...
. Unlike the classical CLT, which requires that the random variables in question have finite
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
and be both
independent and identically distributed
In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usual ...
,
Lindeberg's CLT only requires that they have finite variance, satisfy Lindeberg's condition, and be
independent
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
* Independe ...
. It is named after the Finnish mathematician
Jarl Waldemar Lindeberg.
Statement
Let
be a
probability space
In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
, and
, be ''independent'' random variables defined on that space. Assume the expected values
and variances
exist and are finite. Also let
If this sequence of independent random variables
satisfies Lindeberg's condition:
:
for all
, where 1
is the
indicator function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...
, then the
central limit theorem
In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themsel ...
holds, i.e. the random variables
:
converge in distribution to a
standard normal random variable as
Lindeberg's condition is sufficient, but not in general necessary (i.e. the inverse implication does not hold in general).
However, if the sequence of independent random variables in question satisfies
:
then Lindeberg's condition is both sufficient and necessary, i.e. it holds if and only if the result of central limit theorem holds.
Remarks
Feller's theorem
Feller's theorem can be used as an alternative method to prove that Lindeberg's condition holds.
Letting
and for simplicity
, the theorem states
:if
,
and
converges weakly to a standard
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
as
then
satisfies the Lindeberg's condition.
This theorem can be used to disprove the
central limit theorem
In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themsel ...
holds for
by using
proof by contradiction. This procedure involves proving that Lindeberg's condition fails for
.
Interpretation
Because the Lindeberg condition implies
as
, it guarantees that the contribution of any individual random variable
(
) to the variance
is arbitrarily small, for sufficiently large values of
.
See also
*
Lyapunov condition
*
Central limit theorem
In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themsel ...
References
{{Reflist
Theorems in statistics
Central limit theorem