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 X

_{i}be a random variable sequence that converges to a random variable X with a distribution function F(x) and if Y_{i}is a random variable sequence that converges to a probability of constant c. Then:

- X
_{i}+ Y_{i}is distributed asymptotically as X + c.- X
_{i}Y_{i}is distributed asymptotically as Xc.- X
_{i}/ Y_{i}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 g_{i}, is defined as:

- g
_{1}({x_{n}}) = {x_{n}} - g
_{2}({x_{n}}) = {2x_{n}} - g
_{3}({x_{n}}) = {3x_{n}} - …
- g
_{k}({x_{n}}) = {kx_{n}}

g_{i} converges in probability to a constant c = μ.

And h is defined, in terms of g, as:

h (g_{1}, g_{2}, g_{3}…g_{k}) =

With reference to these two functions, Slutsky’s theorem tells us that the limit of h (g_{1}, g_{2}, g_{3}…g_{k}) 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.

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