# quadratic variation of Brownian motion $B(t)$.

Let $\{X_n\}_{n \geq 1}$ be a sequence of random variables with $\mathbb{E}[X_n] = u$. Suppose $\lim_{n \to \infty}\mathrm{Var}[X_n] = 0$. Do we have that $X_n$ converges to constant $u$ almost surely?

What I ask actually comes from proving the quadratic variation of Brownian motion $B(t)$ is $t$. I was wondering how above argument for $X_n = \sum_i[B(t_i^n)-B(t_{i-1}^n)]^2$ implies that quadratic variation of Brownian motion $B(t)$ is $t$?

• As far as I know, the approximate quadratic variation need not converge a.s. without extra conditions. Refer to this, for instance. I am not sure how you would relate those two problems. – Sangchul Lee Feb 23 '17 at 22:42
• It's a theorem in my textbook. Refer to this books.google.com/… Page 63. – Stupid_Guy Feb 23 '17 at 22:48