1. ## Convergence

If where

1) Show that (convergence in probability)

I no have idea how to develop that

2. ## Re: Convergence

Look, I know that is the second sample moment, and so for rth-mean convergence, I have , and if converges in rth-mean implies that converges in probability.

The other idea is

some of the ideas are right?

3. ## Re: Convergence

This is just a particular example of Weak Law of Large Number, which can be proved typically by the application of Chebyshev's inequality. In this case you can apply this inequality because all (4th) momemts of a normal random variable exists.

4. ## Re: Convergence

Originally Posted by BGM
This is just a particular example of Weak Law of Large Number, which can be proved typically by the application of Chebyshev's inequality. In this case you can apply this inequality because all (4th) momemts of a normal random variable exists.
What did I do wrong?

5. ## Re: Convergence

The first idea is alright - if you already have the result of convergence in mean.
i.e. Are you sure you can directly use the result

without proving it?

For the second idea, the step

is wrong. As a quick check: the RHS is independent of the limit , and as you see there is no random variable inside the probability and it is actually equal to zero.

6. ## Re: Convergence

where

so

This is right?
Anyway it is a particular case of chebyshev's inequality?

8. ## Re: Convergence

Note that when you apply the Markov inequality, it require non-negative random variable. So you cannot apply on the random variable like .

9. ## Re: Convergence

My last try

If and , so

when

So

Is that right? If not please solve for me, because I do not know what to do

10. ## Re: Convergence

bump for help.

11. ## Re: Convergence

Originally Posted by BGM
Note that when you apply the Markov inequality, it require non-negative random variable. So you cannot apply on the random variable like .
You can verify that this is correct?

12. ## Re: Convergence

Almost done. But you should further show that the term indeed converges to 0, since at this point is still depends on

13. ## Re: Convergence

Originally Posted by BGM
Almost done. But you should further show that the term indeed converges to 0, since at this point is still depends on
Thank you, solved.

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