On a functional version of the convergence of a quadratic form in independent martingales to a $\chi^2$ distribution

Let $(a_{ij})_{i,j\ge 1}$ be an infinite matrix of real numbers such that $a_{ii}=0$, $i\ge 1$ and let $(X^i)_{i\ge 1}$ be a sequence of independent martingales such that $\sup_{i\ge1}\mathsf{E}[(X_1^i)^4]<\infty$ and for each $i\ge 1$ the predictable compensator of the quadratic variation of $X^i$ is the identity function. If for each $n\ge 1$, $\sigma_n^2=\sum^n_{i,j=1}a^2_{ij}$ we give a necessary and sufficient condition so that the process defined for each $n\ge 1$ and $t\ge 1$, by $\sigma_n^{-1}\sum_{i<j\le n}a_{ij}X_t^iX^j_t$ converges in law to $((2\sqrt{p})^{-1}\sum_{i=1}^p((B_t^i)^2-t))_{t\le1}$, where $p\ge 1$ and $B^1,\dots,B^p$ are $p$ independent standard Brownian motions. We then study the case where $(X^i)_{i\ge 1}$ is a sequence of independent solutions to the ‘`Structure Equation.’