Let $(\Omega, \mathcal F P)$ be a probability space with a filtration $\mathcal F_t$ and $M = (M_t,t\geq 0)$ be a real-valued continuous $\mathcal F_t$-martingale such that its quadratic variation satisfies $\langle M\rangle_\infty = \infty$ a.s. Then, the strong law of large numbers for martingales yields $$ \lim_{t\to\infty} \frac{M_t}{\langle M\rangle_t}=0, $$ a.s. (see e.g. Lipster & Shirayev Theory of Martingales, Corollary 1.1, p. 144).
I'm looking for a reference for this result, but in the case of vector-valued continuous martingales. In particular, let $M$ be a vector-valued $\mathcal F_t$-martingale (with values in $\mathbb R^d$). In this case, the quadratic variation $\langle M \rangle$ is defined as the unique $Q = (Q_t, t \geq 0)$ with values in $\mathbb R^{d\times d}$ such that $$ M_t \otimes M_t - Q_t $$ is an $\mathcal F_t$-martingale (see e.g. Pavliotis & Stuart, Multiscale Methods: Averaging and Homogenization, Definition 3.18, p. 46).
What can we say in this case? For example, if $\langle M \rangle_t$ is a non-singular (thus in this case symmetric positive definite) matrix a.s. for all $t \geq 0$ and such that $\|\langle M\rangle_\infty\| = \infty$ a.s., can we say that $$ \lim_{t\to\infty} \langle M \rangle_t^{-1} M_t = 0, \qquad \qquad (1) $$ a.s. in $\mathbb R^d$?
Example
I give a simple example of a martingale I am interested in. Let $W$ be a one-dimensional Wiener process with natural filtration $\mathcal F_t$ and let $\beta = (\beta_t, t\geq 0)$ be a $\mathcal F_t$-adapted stochastic process with values in $\mathbb R^d$ and components $\beta_t = (\beta_t^{1}, \ldots, \beta_t^{d})^\top$. Then consider the $\mathcal F_t$-martingale $$ M_t = \int_0^t \beta_s \, \mathrm{d} W_s = \begin{pmatrix}\int_0^t \beta_s^1 \, \mathrm{d} W_s, \ldots, \int_0^t \beta_s^d \, \mathrm{d} W_s \end{pmatrix}^\top. $$ The quadratic variation of $M$ satisfies (by Itô isometry) $$ \langle M \rangle_t=\int_0^t \beta_s \beta_s^\top \, \mathrm{d} s. $$ Can we conclude something in the line of $(1)$?
