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 ...
, Slutsky’s theorem extends some properties of algebraic operations on
convergent sequences
In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numb ...
of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s to sequences of
random variables.
The theorem was named after
Eugen Slutsky
Evgeny "Eugen" Evgenievich Slutsky (russian: Евге́ний Евге́ньевич Слу́цкий; – 10 March 1948) was a Russian and Soviet mathematical statistician, economist and political economist.
Work in economics
Slutsky is princip ...
. Slutsky's theorem is also attributed to
Harald Cramér.
Statement
Let
be sequences of scalar/vector/matrix
random element In probability theory, random element is a generalization of the concept of random variable to more complicated spaces than the simple real line. The concept was introduced by who commented that the “development of probability theory and expansi ...
s.
If
converges in distribution to a random element
and
converges in probability to a constant
, then
*
*
*
provided that ''c'' is invertible,
where
denotes
convergence in distribution
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 t ...
.
Notes:
# The requirement that ''Y
n'' converges to a constant is important — if it were to converge to a non-degenerate random variable, the theorem would be no longer valid. For example, let
and
. The sum
for all values of ''n''. Moreover,
, but
does not converge in distribution to
, where
,
, and
and
are independent.
[See ]
# The theorem remains valid if we replace all convergences in distribution with convergences in probability.
Proof
This theorem follows from the fact that if ''X''
''n'' converges in distribution to ''X'' and ''Y''
''n'' converges in probability to a constant ''c'', then the joint vector (''X''
''n'', ''Y''
''n'') converges in distribution to (''X'', ''c'') (
see here).
Next we apply the
continuous mapping theorem
In probability theory, the continuous mapping theorem states that continuous functions preserve limits even if their arguments are sequences of random variables. A continuous function, in Heine’s definition, is such a function that maps converg ...
, recognizing the functions ''g''(''x'',''y'') = ''x'' + ''y'', ''g''(''x'',''y'') = ''xy'', and ''g''(''x'',''y'') = ''x'' ''y''
−1 are continuous (for the last function to be continuous, ''y'' has to be invertible).
See also
*
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 ...
References
Further reading
*
*
*
{{DEFAULTSORT:Slutsky's Theorem
Asymptotic theory (statistics)
Probability theorems
Theorems in statistics