1. ## Convergence in Distribution

If is a sequence of random variables independent and is Binomial(n,p) with 0<p<1 and either for all n

1) Show that

I know that

But I can not get this result at all, I've tried using characteristic function but also yielded nothing.

2. ## Re: Convergence in Distribution

I'm a little confused about the sequence of random variables

If is a sequence of independent random variables, that means that or ?

3. ## Re: Convergence in Distribution

A sequence of random variables is nothing special; as you can have a sequence of numbers, a sequence of functions etc. for different mathematical objects.

In this question you have typed that all random variables in this sequence have the common distribution (and they are independent). And this is often abbreviated as i.i.d. sequence of random variables. As they have the common distribution, they share the common mean and all other moments / properties as well.

Of course for the sequence is not the same.

4. ## Re: Convergence in Distribution

Originally Posted by BGM
A sequence of random variables is nothing special; as you can have a sequence of numbers, a sequence of functions etc. for different mathematical objects.

In this question you have typed that all random variables in this sequence have the common distribution (and they are independent). And this is often abbreviated as i.i.d. sequence of random variables. As they have the common distribution, they share the common mean and all other moments / properties as well.

Of course for the sequence is not the same.
Thank you, solved.

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