Let $(X_t)_{t \in [0,2\pi]}$ be a continuous Ito martingale. Let $ t_1, \ldots, t_N, t_{N+1} $ be a discretization grid on $[0,2\pi]$. Let s $\in [0,2\pi]$. I would like to show the following convergence in probability: $$ \lim_{N \to \infty} \frac{1}{2\pi} \sum_{n=1}^N F(s-t_n) \, (X_{t_n}-X_{t_{n+1}})^2 \xrightarrow{\mathbb{P}} \frac{d\left[ X, X \right]_s}{ds} $$ where the Fejér kernel is given by $$ F(s-t) = \lim_{K \to \infty} \frac{1}{K} \left( \frac{\sin\left( \frac{K(s - t)}{2} \right)}{\sin\left( \frac{s - t}{2} \right)} \right)^2 $$ I also wonder if we have almost-sure convergence, or path-wise convergence?
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