(The following question arises from my Math.SE question https://math.stackexchange.com/questions/3643865.)

Let $\rho$ be a probability measure on $\mathbb{R} \times (0,\infty)$, and writing $\ \pi_1 \colon \mathbb{R} \times (0,\infty) \to \mathbb{R}\ $ and $\ \pi_2 \colon \mathbb{R} \times (0,\infty) \to (0,\infty)\ $ for the coordinate projections, suppose that $\int \pi_2^2 \, d\rho < \infty$, $\int \pi_1^2 \, d\rho=\int \pi_2 \, d\rho$ and $\int \pi_1 \, d\rho = 0$.

For each $\lambda>0$, consider the one-dimensional stochastic process $(W^{(\lambda)}_t)_{t \geq 0}$ which, for each sample point $\omega$, linearly interpolates the discrete-point-mapping \begin{align*} t_n(\omega) \ \mapsto \ &\frac{1}{\sqrt{\lambda}} \sum_{i=1}^n X_i(\omega) \\ t_n(\omega) \ := \ &\frac{1}{\lambda} \sum_{i=1}^n \Delta_i(\omega) \quad \text{for each $n \geq 0$} \end{align*} where the random vectors $\begin{pmatrix} X_i \\ \Delta_i \end{pmatrix}$, $i \geq 1$, are i.i.d. with law $\rho$.

Equipping $C([0,\infty),\mathbb{R})$ with the topology of uniform convergence on bounded sets, is it the case that the $C([0,\infty),\mathbb{R})$-valued random variable $(W^{(\lambda)}_t)_{t \geq 0}$ converges in distribution to a Wiener process as $\lambda \to \infty$?

The case where $\pi_2$ projects $\rho$ onto a Dirac mass is essentially Donsker's invariance principle; so I am wondering about the more general case. I emphasise that I do *not* wish to assume that $\pi_1$ and $\pi_2$ are independent under $\rho$.

I realise it would probably be good to say a bit more about the motivation behind this question.

The Wiener process has some nice properties, such as increments that are stationary and memoryless. Donsker's theorem describes one way in which a Wiener process can physically arise, namely as a random walk with small step distance $\sqrt{\Delta}$ and high step frequency $\frac{1}{\Delta}$. But as a continuous-time process, this random walk does not have increments that are both stationary and exhibit decay of correlations.

There may be situations in which one wishes to work with a Donsker-like approximation to Brownian motion (e.g. SDEs driven by "bounded noise", to avoid extreme events in the long-term behaviour of the system), but keeping some of the nice properties of Brownian motion, such as increments that are stationary and exhibit decay of correlations. The raw form of Donsker's theorem will not achieve this, and so some modified version such as described above can be used.