I have a problem in finding the asymptotic behavior of this sum: $$\sum_{i=0}^{n-1} \bigl|B^2 (t_{i+1})-B^2 (t_i)\bigr|$$ over $[0,T]$ when $h= t_{i+1}-t_i \to 0$ and $B$ is Brownian motion.
The following is how far I got with this problem. First, I consider:
\begin{align*} \bigl|B^2 (t_{i+1})-B^2 (t_i)\bigr| &=\bigl|[B (t_{i+1})-B (t_i)][B (t_{i+1})+B (t_i)]\bigr|\\ &=\bigl|[B (t_{i+1})-B (t_i)]^2 + 2 B(t_i)[B (t_{i+1})-B (t_i)]\bigr| \\ &\quad<\bigr|[B (t_{i+1})-B (t_i)]^2|+|2 B(t_i)[B (t_{i+1})-B(t_i)]\bigr| \end{align*} I can show that sum of the first element goes to $T$ when $h \to 0$, but I can't find asymptotic behavior of the second element (I tried to show that it will go to infinity). Also, I need to find the lower bound of this in order to show that $\sum_{i=0}^{n-1} \bigl|B^2 (t_{i+1})-B^2 (t_i)\bigr|$ goes to infinity. Did I make a mistake somewhere?