For each $n$ let $M_{n}(t)$ be a martingale on $[0,\infty)$ and $\mathbb{E}(M_n(0))=0$. Also $\sigma(t)\geq 0$ be a function such that $$ [M_n,M_n]_t \xrightarrow{p} \sigma(t), \,\,\&\,\, \sup_{n}\mathbb{E}([M_n,M_n]_t)<\infty, $$ as $n\rightarrow \infty$, where $[M_n,M_n]_t =\sum_{0<s\leq t}(M_n(s)-M_n(s-))^2$. Furthermore assume that $$ M_n(0)\xrightarrow{d} N(0,\sigma^2), $$ as $n \rightarrow \infty$. Can we prove that $M_n\xrightarrow{d}M$, in $D[0,\infty)$ where $M(t)$ is a Gaussian martingale with $\mathbb{E}(M(t))=0$ and $\mathbb{V}(M(t))= \sigma(t)+\sigma^2$? (we can assume all moments of $M_n$ are uniformly bounded). Otherwise, can the limit $M$ be different (or does not exists)?

Note: If $M_n(0)=0$ then the assertion holds with $\sigma=0$ (Proposition 9.1 in ``Functional limit theorems for multitype branching processes and generalized Pólya urn" by Svante Janson).

For each $n$ let $M_{n}(t)$ be a martingale on $[0,\infty)$ and $\mathbb{E}(M_n(0))=0$. Also $\sigma(t)\geq 0$ be a continuous function such that $$ [M_n,M_n]_t \xrightarrow{p} \sigma(t), \,\,\&\,\, \sup_{n}\mathbb{E}([M_n,M_n]_t)<\infty, $$ as $n\rightarrow \infty$, where $[M_n,M_n]_t =\sum_{0<s\leq t}(M_n(s)-M_n(s-))^2$. Furthermore assume that $$ M_n(0)\xrightarrow{d} N(0,\sigma^2), $$ as $n \rightarrow \infty$. Can we prove that $M_n\xrightarrow{d}M$, in $D[0,\infty)$ where $M(t)$ is a Gaussian martingale with $\mathbb{E}(M(t))=0$ and $\mathbb{V}(M(t))= \sigma(t)+\sigma^2$? (we can assume all moments of $M_n$ are uniformly bounded). Otherwise, can the limit $M$ be different (or does not exists)?

Note: If $M_n(0)=0$ then the assertion holds with $\sigma=0$ (Proposition 9.1 in ``Functional limit theorems for multitype branching processes and generalized Pólya urn" by Svante Janson).