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




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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 $\rho$.



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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 $\rho$.




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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.



Is there a generalised version of the Donsker invariance principle for a "sort-of continuous-time-random-walk"?
I think Iosif Pinelis is correct, but his comment should be expanded as follows.




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Notation: let
$$ p(t) = \lfloor t + 1\rfloor - t , \qquad q(t) = t - \lfloor t\rfloor . $$
Whenever we have a discrete-time process $Z\_n$, we extend it into a continuous one, piecewise linear, defined by:
$$ Z\_t = p(t) Z\_{\lfloor t\rfloor} + q(t) Z\_{\lfloor t + 1\rfloor} . $$
With no loss of generality assume that $\mathbb{E} \Delta\_n = 1$ to simplify the notation.




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**Step 1.** The bi-variate process $$(Y\_n,Z\_n) = (\sum\_{i=1}^n (\Delta\_i - 1), \sum\_{i=1}^n X\_i)$$ satisfies Donsker's invariance principle, that is,
$$ (Y^{(\lambda)}\_t, Z^{(\lambda)}\_t) = (\lambda^{-1/2} Y\_{\lambda t}, \lambda^{-1/2} Z\_{\lambda t}) $$
converges weakly (in the Banach space of continuous functions on $[0,T]$, for any given $T > 0$) to a two-dimensional Brownian motion $(\tilde Y\_t, \tilde Z\_t)$.




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**Step 2.** Let $$(T\_n,Z\_n) = (\sum\_{i=1}^n \Delta\_i, \sum\_{i=1}^n X\_i) = (Y\_n + n, Z\_n) $$ and consider
$$ (T^{(\lambda)}\_t, Z^{(\lambda)}\_t) = (\lambda^{-1} T\_{\lambda t}, \lambda^{-1/2} Z\_{\lambda t}) = (\lambda^{-1/2} Y^{(\lambda)}\_t + t , Z^{(\lambda)}\_t) .$$
It is then easy to see that $(T^{(\lambda)}\_t, Z^{(\lambda)}\_t)$ converges weakly (in the same sense) to $$(\tilde T\_t, \tilde Z\_t) = (t, \tilde Z\_t),$$ where $\tilde Z\_t$ is the Brownian motion.


This is somewhat technical, but relatively standard, I think: one can use the characterisation of weak convergence of processes in terms of convergence of finite-dimensional distributions and equicontinuity (or $J$-compactness if one prefers the more general approach via Shorokhod topology). See Theorem 1.6.2 in [[Silvestrov D.S. (2004) *Weak convergence of stochastic processes*. In: *Limit Theorems for Randomly Stopped Stochastic Processes. Probability and its Applications*. Springer, London]](https://link.springer.com/content/pdf/10.1007%2F978-0-85729-390-9_1.pdf) for details.




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**Step 3.** The process $W\_t^{(\lambda)}$ is a continuous, piecewise linear process defined by
$$ W^{(\lambda)}\_t = Z^{(\lambda)}\_{(T^{(\lambda)})^{-1}\_t} .$$
Our claim is: this process converges weakly
$$ \tilde W\_t = \tilde Z\_{\tilde T^{-1}\_t} = \tilde Z\_t. $$


Here we use the very definition of weak convergence. Let $\Phi$ be a continuous functional on the space of paths $\omega\_t$ of $W^{(\lambda)}\_t$ or $\tilde W\_t$. Then $\Phi$ induces a functional $\Psi$ on the space of paths $(\tau\_t, \zeta\_t)$ of $(T^{(\lambda)}\_t, Z^{(\lambda)}\_t)$ and $(\tilde T\_t, \tilde Z\_t)$, in the following sense:
$$ \Psi((\tau\_t), (\zeta\_t)) = \Phi((\tau\_{\zeta^{-1}\_t})) , $$
where $\zeta^{-1}\_t = \inf \{ s : \tau\_s \ge t \}$ is the generalised inverse of $\zeta\_t$.


Of course $\Psi$ is no longer continuous. However, it is easy to see that the set of discontinuities of $\Psi$ has zero measure with respect to the law of $(\tilde T\_t, \tilde Z\_t)$. Roughly speaking: the paths of $(\tilde T\_t, \tilde Z\_t)$ are of the form $((\tilde \tau\_t), (\tilde \zeta\_t))$, where $\tau\_t = t$ for all $t$. Given such a path and $\varepsilon > 0$, choose $\delta > 0$ so that the $\delta$-modulus of continuity of $\tilde \zeta\_t$ is less than $\varepsilon$. If $(\tau\_t)$ is $\delta$-close to $(\tilde \tau\_t) = (t)$ and $(\zeta\_t)$ is $\varepsilon$-close to $(\tilde \zeta\_t)$, then $|\tau\_t^{-1} - t| < \delta$ (by a simple calculus exercise), and hence
$$ |\omega\_t - \tilde \omega\_t| = |\zeta\_{\tau\_t^{-1}} - \tilde \zeta\_t| \leqslant |\zeta\_{\tau\_t^{-1}} - \tilde \zeta\_{\tau\_t^{-1}}| + |\tilde Z\_{\tau\_t^{-1}} - \tilde Z\_t| \le 2 \varepsilon . $$
This proves continuity of $\Psi$ at $((\tilde \tau\_t), (\tilde \zeta\_t))$, as desired.


This is perfectly sufficient for our needs: weak convergence of $(T^{(\lambda)}\_t, Z^{(\lambda)}\_t)$ to $(\tilde T\_t, \tilde Z\_t)$ implies that
$$ \mathbb{E} \Psi((T^{(\lambda)}\_t), (Z^{(\lambda)}\_t)) = \mathbb{E} \Phi((W^{(\lambda)}\_t)) $$
converges to
$$ \mathbb{E} \Psi((\tilde T\_t), (\tilde Z\_t)) = \mathbb{E} \Phi((\tilde W\_t) , $$
and consequently $W^{(\lambda)}\_t$ converges weakly to $\tilde W\_t$.




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I am sure the above argument is well-known, for instance, in the literature on "Lévy flights", "continuous-time random walks", or similar. There are still some details missing, but I hope the idea is now clear.


