Given a sequence $\Pi=\{\pi_n\}$ of partitions of an interval $[0,T]$ the quadratic variation of a path $x\colon [0,T]\to \mathbb{R}$ is defined by $$ [x]=\lim_{n\to +\infty}\sum_{\pi_n}|x(t_{i+1})-x(t_i)|^2. $$

I am interesting in any result which provides conditions under which a uniform convergence $x_m\to x$ implies convergence in quadratic variation: $[x_m-x]\to 0$. As well as the other direction: when does convergence in quadratic variation: $[x_m-x]\to 0$ imply pointwise convergence $x_m\to x$.

Obviously, in general this implications are false. It is known that for any continuous path $x$ with positive probability there are Brownian paths in arbitrary small neighborhood of $x$. Quadratic variation of a path of a Brownian motion is almost surely equal to $T$, but we can choose $x$ with $[x]\neq T$.

For the second implication an obvious counterexample would be $x_m(t)=m t,\, x(t)=0.$

I am interested if there are results proving those implications under additional requirements on $x_m, x$.


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