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 ...

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

X_n, Y_n

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

X_n

converges in distribution to a random element

X

and

Y_n

converges in probability to a constant

c

, then *

X_n + Y_n \ \xrightarrow\ X + c ;

X_nY_n \ \xrightarrow\ Xc ;

X_n/Y_n \ \xrightarrow\ X/c,

provided that ''c'' is invertible, where

\xrightarrow

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converges to a constant is important — if it were to converge to a non-degenerate random variable, the theorem would be no longer valid.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

X_n \sim (0,1)

and

Y_n = -X_n

. The sum

X_n + Y_n = 0

for all values of ''n''. Moreover,

Y_n \, \xrightarrow \, (-1,0)

, but

X_n + Y_n

does not converge in distribution to

X + Y

, where

X \sim (0,1)

Y \sim (-1,0)

, and

X

and

Y

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).

Statement

Proof

See also

References

Further reading