Approximating functions of Brownian motion variation

Question

Consider the unit interval $[0,1]$ and the equi-spaced partition $t_j=\frac{j}{n}$, with $j=0,\dots,n$. Suppose that $\sigma_t$ is a positive bounded cadlag process and $W_t$ a Brownian motion. I want to do the following approximation $$ g\left(\sqrt{n}\,\sigma_{\left(j-1\right)/n}\,\left(W_{(j+1)/n}-W_{j/n}\right)\right) = g\left(\sqrt{n}\,\sigma_{\left(j-2\right)/n}\,(W_{j/n}-W_{(j-1)/n})\right)+o_P(n^{-\alpha}) $$ where $g$ is a regular enough function and $\alpha>0$. I'v used "regular enough" because in some papers the above approximation is claimed to hold for any function with at most polynomial growth. The idea is that on the left-hand side we have a $\mathcal{F}_{(j+1)/n}$-measurable quantity which is approximated by shifting the time backward of one unit, giving a $\mathcal{F}_{j/n}$-measurable, plus an error. Any idea is welcome.