I'm currently revising for a probability course and I just came across the following lemma
Let $(B_t)_{t\geq 0}$ be a one-dimensional Brownian motion and $0=t_0^n<...<t_{p_n}^n=t$ be a subdivison of $[0,t]$ for some $t>0$ such that $\sup_{1\leq i \leq p_n}(t_i^n-t_{i-1}^n)\to 0$ when $n\to \infty$. Then we have that $\lim_{n\to\infty}\sum_{i=1}^{p_n}(W_{t_i^n}-W_{t_{i-1}^n})^2=t$ ind $L^2$. Moreover, if $\sum_{n\geq 1}\sum_{i=1}^n (t_i^n-t_{i-1}^n)^2$ is finite, then the convergence also holds almost surely
My problem is that I don't see, why we need this extra assumption to get a.s. convergence. When I asked the assistent, he just told me that there are counter examples but he didn't tell me which ones. So can anyone help me and post a counter example or another explanation why a.s. convergence won't hold without this extra assumption?
Thanks a lot!!
For a subdivision $\Pi=\{0=t_0<t_1<\cdots<t_p=t\}$ of $[0,t]$, let $Q(\Pi)$ denote the corresponding quadratic variation $\sum_{k=1}^{p}(B_{t_k}-B_{t_{k-1}})^2$ of the Brownian motion $(B_t)_{t\ge 0}$. It was shown by Paul Lévy in 1940 that for almost every sample path of $(B_t)$, $\sup\{Q(\Pi):|\Pi|<\delta\}=\infty$. Here $|\Pi|$ is the maximum separation between successive points of $\Pi$. This indicates that some care will be needed in deducing an almost sure convergence result for $Q(\Pi_n)$ for a sequence of partitions $\{\Pi_n\}_{n\ge 1}$.
On the other hand, if the sequence $\{\Pi_n\}$ is nested and $\lim_n|\Pi_n|=0$, then $\lim_nQ(\Pi_n)=t$ almost surely; this is also due to Lévy (1940), who observed that $n\mapsto Q(\Pi_n)$ is a (backward) martingale.