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Slutsky’s Theorem

and Continuous Mapping Theorem

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We have studied about different modes of convergence in this article. In this post, I will extend those concepts one step further and discuss about Slutsky’s theorem.

Let’s see where is it needed?

If there are several limits and we want to combine them in multiple ways e.g. sum of limits, multiplication etc. All these are possible till we have convergence almost surely and convergence in probability. But if one of the random variables converges in distribution, then we need another stronger convergence to be able to perform such operations on them.

This sets the context of what we will be discussing in this article.

Assume, we have two sequences of random variables Xₙ and Yₙ which converge to random variables X and Y.

Two of the convergences i.e. almost surely (a.s.) and convergence in probability are preserved as follows:

But such rules and operations do not apply on convergence in distribution. Then what tool we have in such case.

Yes, you guessed it right. Its Slutsky’s theorem which states the properties of algebraic operations about the convergence of random variables.

As explained here, if Xₙ converges in distribution to a random element X and Yₙ converges in probability to a constant c, i.e.

Note the convergence of Yn to a constant c
By Slutsky’s Theorem

Example of Slutsky’s theorem:

Assuming a random variable X has mean 𝜇 and standard deviation 𝜎, then by Central Limit Theorem:

Central Limit Theorem

Next, let’s define t-statistic as below:

Then, by Slutsky’s Theorem:

Explanation: As the individual limits converge in distribution and probability to standard normal and 1 respectively, then by Slutsky’s theorem, the product of such limits converges in distribution to their product i.e. N(0,1).

Note that Slutsky’s theorem’s proof is largely dependent on another important theorem i.e. Continuous Mapping Theorem, and that’s what we will discuss next.

Continuous Mapping Theorem:

Before understanding the theorem, let’s see what a continuous function is.

A continuous function is a function that maps convergent sequences into convergent sequences:

Continuous Function

Put differently, a continuous function can be represented as a single unbroken curve (or hyperplane in many dimensions) for a given domain. For example, f(x) = 1/x is continuous when D₊: R>0, but not for domain over R as the function is not defined at x = 0

Similarly, the continuous mapping theorem suggests that above will still hold true if

  • deterministic sequence {xₙ} is replaced with a sequence of random variables {Xₙ}
  • standard notion of convergence of real numbers is replaced with that of convergence of random variables
Continuous Mapping Theorem

Primarily, Continuous Mapping theorem helps us to find out that if Xₙ converges to X in some sense, then does g(Xₙ) also converges to g(X) in the same sense.

In this article, we picked up the basics of convergence of random variables from previous article and studied about the Slutsky’s theorem and Continuous Mapping theorem.

Thanks for reading. Happy Learning!!!




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