Suppose we have a set $M$ of random variables on a probability space. Then we defined boundedness of $M$ as,
$M$ is bounded if $\sup_{X\in M}P(|X|>N) \to 0$ as $ N \to \infty$.
This definition means that the measure of the set where the elements of $ M $ are big is very small, in fact tends to zero. I have three questions to this definition and some further conclusions.
Now suppose we have a unbounded set. My questions are:
- Unbounded would mean, that for every $N>0$ there exists an $ \epsilon >0 $ and a $ X \in M $ such that $ P(|X| > N) \ge \epsilon $. Is this conclusion right?
- If theres a sequence $ (X_n) $ of random variables, unbounded and positive, then there's a subsequence $ (X_{n_k}) $ and a $\lambda>0 $ such that $ P(|(X_{n_k})| > k)\ge \lambda $ for every $ k \in \mathbb{N}$.
My observations so far to 2. After the comment of Srivatsan (see below), we therefore have: \exists \epsilon > 0 sucht that for all N>0 exists a $ X_n $ such that P(X_n(\omega) > N) \ge \epsilon Put $ N=1 $, hence there is a $ X_{n_1} $ sucht that P(X_{n_1} > 1 ) \ge \epsilon. Now put $ N=2 $, hence there is a $ X_i $ such that $ P(X_i > 2) \ge \epsilon$. Now the problem is, why do I know that $ i> n_1 $ ? Otherwise it isn't a subsequence.
Thanks for your help
hulik