# Convergence of Random Variables on Discrete Prob Spaces

• IniquiTrance
In summary, weak convergence and almost sure convergence are the same in discrete probability spaces. This means that not only does almost sure convergence imply weak convergence, but also weak convergence implies almost sure convergence. However, this may not be true in all cases and a counterexample has been provided. In order to prove this in discrete spaces, one could assume that a sequence does not almost surely converge and show that this happens on at least one discrete event with non-zero probability, preventing the sequence from weak convergence.
IniquiTrance
Well, I thought I understood the difference between (weak) convergence in probability, and almost sure convergence.

My prof stated that when dealing with discrete probability spaces, both forms of convergence are the same.

That is, not only does A.S. convergence imply weak convergence, as it always does, but in the discrete case, weak convergence implies A.S. convergence.

I've been trying to wrap my head around why this is so, but can't seem to "see" it.

Any ideas?

Thanks!

I don't think it's true that weak convergence implies a.s. conv. in the discrete case.

@bpet:
Thanks for that example in that thread. Like I said, I thought I finally understood the difference.

Yet my professor said one can prove that on a discrete probability space:

$$X_n(\omega)\stackrel{p}{\longrightarrow} X(\omega)\implies X_n(\omega)\stackrel{A.S}{\longrightarrow} X(\omega)$$

This is a totally different question!

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Eynstone said:
I don't think it's true that weak convergence implies a.s. conv. in the discrete case.

I don't either, yet my prof said it can be proven that this is true... I can't see how though...

IniquiTrance said:
@bpet:
Thanks for that example in that thread. Like I said, I thought I finally understood the difference.

Yet my professor said one can prove that on a discrete probability space:

$$X_n(\omega)\stackrel{p}{\longrightarrow} X(\omega)\implies X_n(\omega)\stackrel{A.S}{\longrightarrow} X(\omega)$$

This is a totally different question!

Ok sorry I didn't take into account that, even though the individual archery outcomes are discrete, there isn't necessarily a discrete event space underlying the joint distribution of the infinite sequence.

An approach for the discrete space could be to assume that a sequence does not a.s. converge and show that this happens on at least one discrete event with non-zero probability (because every non-zero probability contains at least one atom), and this prevents the sequence from weak convergence.

HTH

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## 1. What is meant by convergence of random variables on discrete probability spaces?

The convergence of random variables on discrete probability spaces refers to the behavior of a sequence of random variables as the number of trials or experiments increases towards infinity. It is a measure of how closely the sequence of random variables approaches a certain value or limit.

## 2. How is convergence of random variables on discrete probability spaces different from convergence on continuous probability spaces?

The main difference between convergence on discrete and continuous probability spaces is that in discrete spaces, the random variables can only take on a finite or countably infinite number of values, whereas in continuous spaces, the random variables can take on any value within a certain range. This difference affects the methods used to determine convergence and the interpretation of the results.

## 3. What are some examples of discrete probability spaces?

Examples of discrete probability spaces include coin tosses, rolling a die, and drawing cards from a deck. In these cases, the random variables represent the possible outcomes of the experiments and their probabilities can be calculated based on the number of favorable outcomes divided by the total number of outcomes.

## 4. What are some common methods for determining convergence of random variables on discrete probability spaces?

Common methods for determining convergence on discrete probability spaces include the law of large numbers, which states that as the number of trials increases, the average of the random variables will approach the expected value, and the central limit theorem, which states that the sum of a large number of independent and identically distributed random variables will approach a normal distribution.

## 5. Why is the convergence of random variables on discrete probability spaces important in scientific research?

Convergence of random variables on discrete probability spaces is important in scientific research because it allows for the prediction and analysis of outcomes in experiments with a finite number of possible outcomes. It also provides a framework for understanding the behavior of random variables and making inferences about the underlying processes that generate the data.

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