Slutsky’s theorem is used to explore convergence in probability distributions. It tells us that if a sequence of random vectors converges in distribution and another sequence converges in probability to a constant, those sequences are jointly convergent in distribution. Basically, it allows you to use convergence results proved for one sequence for other closely related sequences.
Slutsky’s has no real practical applications; Its use is mostly limited to theoretical mathematical statistics (specifically, asymptotic theory). For example, it extends the usefulness of the Central Limit Theorem. Other uses include:
- Explore convergence in functions of random variables.
- Highlight critical properties of converging random variables.
- Calculate the convergence of any continuous function of a set of statistics, providing that one set of those statistics converges (Davidson, 1994).
Formal Definition of Slutsky’s Theorem
More formally, Manoukian (1986) defines Slutsky’s theorem as follows:
If Xi be a random variable sequence that converges to a random variable X with a distribution function F(x) and if Yi is a random variable sequence that converges to a probability of constant c. Then:
- Xi + Yi is distributed asymptotically as X + c.
- Xi Yi is distributed asymptotically as Xc.
- Xi / Yi is distributed asymptotically as X / c for c ≠ 0.
The theorem can also be written more succinctly as (from Proschan & Shaw, 2016):
Suppose that Xn→DX, An↠pA, and Bn↠pB,
where A and B are constants. Then AnXn+Bn↠DAX+B.
Simple Example
First, we need to define a couple of functions, g and h. The family of functions gi, is defined as:
- g1({xn}) = {xn}
- g2({xn}) = {2xn}
- g3({xn}) = {3xn}
- …
- gk({xn}) = {kxn}
gi converges in probability to a constant c = μ.
And h is defined, in terms of g, as:
h (g1, g2, g3…gk) =
With reference to these two functions, Slutsky’s theorem tells us that the limit of h (g1, g2, g3…gk) as n approaches infinity is:
(k(k + 1) / 2) · μ
(Adapted from Kapadia et.al)
References:
Davidson, J. (1994). Stochastic Limit Theory: An Introduction for Econometricians.
Kapadia et. al. (2005). Mathematical Statistics With Applications.
Manoukian (1986) Mathematical Nonparametric Statistics. CRC Press.
Proschan, M. & Shaw, P. (2016). Essentials of Probability Theory for Statisticians. CRC Press.
Stephanie Glen. "Slutsky’s Theorem: Definition" From StatisticsHowTo.com: Elementary Statistics for the rest of us! https://www.statisticshowto.com/slutskys-theorem/
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