Slutsky's theorem

In mathematics, in particular probability theory, Slutsky's theorem [ cite book
last = Grimmett
first = G.
coauthors = Stirzaker, D.
title = Probability and Random Processes
year = 2001
publisher = Oxford
pages = 3rd ed., exercise 7.2.5 ] , named after Eugen Slutsky [Slutsky, E.,Über stochastische Asymptoten und Grenzwerte. (German)Metron 5, Nr. 3, 3-89 (1925). [http://www.zentralblatt-math.org/zmath/en/search/?q=an:51.0380.03&format=complete JFM 51.0380.03] ] , extends some properties of algebraic operations on convergent sequences of real numbers to sequences of random variables. The related mapping theorem extends the theorem on a continuous mapping of a convergent sequence of real numbers to a continuous mapping of a sequence of random variables.

tatement of Slutsky's theorem

The symbol $xrightarrow\{mathcal\; D\}$ stands for convergence in distribution.

Let $(X\_n)$ and $(Y\_n)$ be sequences of univariate random variables. If:

::$X\_n\; ,\; xrightarrow\{mathcal\; D\}\; ,\; X$::$Y\_n\; ,\; xrightarrow\{mathcal\; D\}\; ,\; c$

where $c$ is a constant, then:

::$X\_n\; +\; Y\_n,\; xrightarrow\{mathcal\; D\}\; ,\; X\; +\; c$::$X\_nY\_n\; ,\; xrightarrow\{mathcal\; D\}\; ,\; cX.$

The continuous mapping theorem

If $X\_n$ are random elements with values in a metric space and $X\_n\; xrightarrow\{mathcal\; D\}\; ,\; X$, $h$ is a function on the metric space, and the probability that $X$ attains a value where $h$ is discontinuous is zero, then $h(X\_n)xrightarrow\{mathcal\; D\}h(X)$ (cite book
last = Billingsley
first = Patrick
title = Convergence of Probability Measures
year = 1969
publisher = John Wiley & Sons ISBN 0471072427] page 31, Corollary 1, [cite book
last = Billingsley
first = Patrick
title = Convergence of Probability Measures
year = 1999
publisher = John Wiley & Sons
pages = 2nd edition ISBN 0471197459] page 21, Theorem 2.7)

This includes for example the convergence of the sum of two sequences of random variables $X\_n$ and $Y\_n$ (the random element is the pair of the random variables, the continuous function is the mapping of the pair to the result of the operation), but only in the case where ::$(X\_n,\; Y\_n)\; ,\; xrightarrow\{mathcal\; D\}\; ,\; (X,Y)$. We note that this does not lead to a more general case of Slutsky's Theorem, because that would require only the assumption::$X\_n\; ,\; xrightarrow\{mathcal\; D\}\; ,\; X$ and $Y\_n\; ,\; xrightarrow\{mathcal\; D\}\; ,\; Y$,which does not imply $(X\_n,\; Y\_n)\; ,\; xrightarrow\{mathcal\; D\}\; ,\; (X,Y)$, so we cannot apply the Continuous mapping theorem.

Convergence to a constant

The following is a corollary of the mapping theorem for convergence in probability to a constant ( page 31, Corollary 2). For a rational function "h", this is also called Slutsky's theorem ( page 34):

If $X\_n$ are random elements with values in a metric space and $X\_n\; xrightarrow\{P\}\; ,\; a$, $h$ is a function on the metric space, and $h$ is continuous at $a$, then $h(X\_n)\; xrightarrow\{P\}\; ,\; h(a)$.

References