Slutsky's theorem

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In mathematics, in particular probability theory, Slutsky's theorem[1], named after Eugen Slutsky[2], 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.

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[edit] Statement of Slutsky's theorem

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

Let (Xn) and (Yn) 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.

[edit] The continuous mapping theorem

If Xn 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) ([3] page 31, Corollary 1, [4] page 21, Theorem 2.7)

This includes for example the convergence of the sum of two sequences of random variables Xn and Yn (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.

[edit] Convergence to a constant

The following is a corollary of the mapping theorem for convergence in probability to a constant ([3] page 31, Corollary 2). For a rational function h, this is also called Slutsky's theorem ([3] page 34):

If Xn 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).

[edit] References

  1. ^ Grimmett, G.; Stirzaker, D. (2001). Probability and Random Processes. Oxford, 3rd ed., exercise 7.2.5. 
  2. ^ Slutsky, E., Über stochastische Asymptoten und Grenzwerte. (German) Metron 5, Nr. 3, 3-89 (1925). JFM 51.0380.03
  3. ^ a b c Billingsley, Patrick (1969). Convergence of Probability Measures. John Wiley & Sons.  ISBN 0471072427
  4. ^ Billingsley, Patrick (1999). Convergence of Probability Measures. John Wiley & Sons, 2nd edition.  ISBN 0471197459