1. ## mean square limit

I'm spending hours trying to develop and prove the mean-square-limit.

Does anyone know where I could find an exhaustive and detailed proof online?

I would really appreciate!

Here is the concept:

Assume the finite time t is sliced into n identical partitions.
Define tj as the time up to the end of the j-th partition.

$\displaystyle t_j= \frac{jt}{n}$

The mean-square-limit sais that
$\displaystyle \lim_{n \rightarrow \infty} E\left[ \left( \sum_{j=1}^n\ (X(t_j)-X(t_{j-1}))^2-t \right)^2 \right] = 0$

where X's are brownian motions and E is the expectation.

Limit is zero because (developing and resolving) E=O(1/n)

2. Originally Posted by paolopiace
I'm spending hours trying to develop and prove the mean-square-limit.

Does anyone know where I could find an exhaustive and detailed proof online?

I would really appreciate!

Here is the concept:

Assume the finite time t is sliced into n identical partitions.
Define tj as the time up to the end of the j-th partition.

$\displaystyle t_j= \frac{jt}{n}$

The mean-square-limit sais that
$\displaystyle \lim_{n \rightarrow \infty} E[(\sum_{j=1}^n\ (X(t_j)-X(t_{j-1}))^2-t)^2] = 0$

where X's are brownian motions and E is the expectation.

Limit is zero because (developing and resolving) E=O(1/n)
Check your brackets, they don't match.

RonL

3. Originally Posted by CaptainBlack
Check your brackets, they don't match.

RonL
Brackets are correct...
It's the E of the {square of the [sum of the (deltas squared) minus time]}