1. ## Convergence in probability

If and

Show that

What I think:
, so

now

doing
assuming and are independent.
now applying chebyshev's

Finally

2. ## Re: Convergence in probability

I guess you cannot assume and are independent. One crucial thing here which you may missed is that

after some simplifications.

3. ## Re: Convergence in probability

Originally Posted by BGM
I guess you cannot assume and are independent. One crucial thing here which you may missed is that

after some simplifications.
I cannot see how

4. ## Re: Convergence in probability

I cannot see how
I'll provide you some assurance that the equality in reference does in fact hold. Write everything out in summation form, expand it all out, cancel the terms that you can and things will simplify. One thing to keep in mind

it's just a very simple change to the the definition of but it's useful to keep in mind when dealing with these summation problems.

5. ## Re: Convergence in probability

Originally Posted by Dason
I'll provide you some assurance that the equality in reference does in fact hold. Write everything out in summation form, expand it all out, cancel the terms that you can and things will simplify. One thing to keep in mind

it's just a very simple change to the the definition of but it's useful to keep in mind when dealing with these summation problems.
Yeah, I can see it now. But I wonder if the way applied to Chebyshev's inequality is correct.

Man I'm really confused now
or

6. ## Re: Convergence in probability

Originally Posted by BGM
I guess you cannot assume and are independent. One crucial thing here which you may missed is that

after some simplifications.
Anyway, is right?

7. ## Re: Convergence in probability

The Chebyshev inequality is:

where . In particular we put , then we have

From your work I think you have applied it correctly. The spirit of this useful bound is that if you want to show a sequence of random variable is converging towards to its common mean, then then you just need to show the variance is converging to zero.

The tricky part here is that

1. the calculation of the variance can be quite tedious
2. the mean of is a sequence of , converging to but not exactly equal to ( is asymptotically unbiased, but biased estimator of ). So you may need to have a little adjustment before applying it.

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