Assume you are given a sequence of random variables $(X_i)_{i\geq1}$. Assume moreover that they are sufficiently smooth, say $\mathbb E[X^2]<+\infty$.
Define the diffusion-scaled partial sum as
\begin{equation} \hat S_n(t) = \frac{1}{\sqrt{n}}\left(S( nt) - \mathbb E[X]nt\right), \end{equation} where $S(t) = \sum_{i=1}^{\lfloor t\rfloor} X_i$, and the diffusion-scaled counting process as \begin{equation} \hat N_n(t) = \frac{1}{\sqrt{n}}\left(N( nt) - \frac{1}{\mathbb E[S]}nt\right), \end{equation} where $N(t) = \max\{k\in\mathbb N_0: S( k)\leq t\}$.
It is well known that $\hat S_n(t)$ converges to some stochastic process $S(t)$ (in distribution in the Skorokhod space endowed, say, with the $J_1$ topology) if and only if $\hat N_n(t)$ converges to $\frac{1}{\mathbb E[S]}S\left(\frac{t}{\mathbb E[S]}\right)$.
My question is the following: is there a standard argument, or reference, showing that the above result can be extended to show that
\begin{equation} d(\hat S_n(t), \mathbb E[S]\hat N_n(\mathbb E[S]t))\stackrel{\mathbb P}{\rightarrow}0, \end{equation} where $d(\cdot,\cdot)$ is an appropriate metric? In other words, does there exist a pathwise version of the above result?