probability theory Probability theory or probability calculus 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 expre ...

, Slutsky's theorem extends some properties of algebraic operations on convergent sequences 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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

s to sequences of

random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...

s. The theorem was named after Eugen Slutsky. Slutsky's theorem is also attributed to

Harald Cramér Harald Cramér (; 25 September 1893 – 5 October 1985) was a Swedish mathematician, actuary, and statistician, specializing in mathematical statistics and probabilistic number theory. John Kingman described him as "one of the giants of statis ...

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 expansio ...

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 sequences of random variables, including ''convergence in probability'', ''convergence in distribution'', and ''almost sure convergence''. The different notions of conve ...

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to a constant is important — if it were to converge to a non-degenerate random variable, the theorem would be no longer valid.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

, 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