Can a Wiener process be obtained as the limit of a "memoryless collision time" model?

Question

Let $(N_t)_{t \geq 0}$ be a Poisson process of intensity $1$, and for each $\lambda>0$ and $t \geq 0$ let $$ W^{(\lambda)}_t = \sqrt{\lambda} \int_0^t (-1)^{N_{\lambda s}} \, ds = \frac{1}{\sqrt{\lambda}} \int_0^{\lambda t} (-1)^{N_s} \, ds. $$

Is it the case that for each $T>0$, as $\lambda \to \infty$ the law of the $C([0,T],\mathbb{R})$-valued random variable $(W^{(\lambda)}_t)_{t \in [0,T]}$ converges weakly to the Wiener measure on $C([0,T],\mathbb{R})$?

(Here, $C([0,T],\mathbb{R})$ is equipped with the topology of uniform convergence.)

Remark. If the answer is yes, then this may be a more physically intuitive way of thinking about Wiener processes than as the limit of a simple random walk: a model of random back-and-forth collision times "feels more physically motivated" (particularly when trying to visualise physical Brownian motion of particles) than a random decision to move either left or right at every time-step of a seemingly arbitrary fixed duration. It would also be a good way of formalising the notion that ("unbounded") one-dimensional Gaussian white noise can be obtained as a limit of ("bounded") dichotomous Markov noise.

If the answer to the question is yes, then this seems like a very basic fact; are there any references with this fact (either as a theorem or an exercise)?

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My very crude intuition for a positive answer:

It "feels obvious" that for large $\lambda$, the stochastic process $(W_t^{(\lambda)})_{t \geq 0}$ has "approximately" stationary independent increments, and $\mathbb{E}[W_t^{(\lambda)}] \approx 0$ for all $t \geq 0$.

So now, for fixed $\tau>0$, let us consider the shape and variance of the distribution of $W_\tau^{(\lambda)}$. For each $n \in \mathbb{N}$, let $T_n=\inf\{t>0:N_{\lambda t}=n\}$. Since the random variables $T_i-T_{i-1}$ are independent and exponentially distributed with variance $\frac{1}{\lambda^2}$, the variance of $W_{T_n}^{(\lambda)}$ is $\frac{n}{\lambda}$, and applying the central limit theorem to $T_1 + (T_3-T_2) + (T_5-T_4) + \ldots$ and to $(T_2-T_1) + (T_4-T_3) + (T_6-T_5) + \ldots$ gives that for large $n$, the distribution of $W_{T_n}^{(\lambda)}$ is approximately normal in shape. Hence the distribution of $Y:=W_{T_{\lfloor \lambda\tau \rfloor}}^{(\lambda)}$ is approximately a normal distribution with a variance of approximately $\tau$. Now if we fix a small $\varepsilon>0$, writing $I_\varepsilon$ for the stochastic interval $$ I_\varepsilon = [T_{\lfloor \lambda(\tau - \varepsilon) \rfloor},T_{\lfloor \lambda(\tau + \varepsilon) \rfloor}], $$ taking sufficiently large $\lambda$ should give that

• $\tau \in I_\varepsilon$ with high probability, and
• the random variable $\max_{t \in I_\varepsilon} |W_t^{(\lambda)}-Y|$ is close to $0$ with high probability.

Hence it seems intuitive that the law of $W_\tau^{(\lambda)}$ should be approximately equal to the law of $Y$, and so in particular, should be approximately normally distributed with approximate variance $\tau$.

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I've now posted a related more general question at MathOverflow, https://mathoverflow.net/questions/360363. If the answer to that question is yes, then the answer to this one should be as well (by the argument of zhoraster using the generalised invariance principle asked about in that question).

Answer 1

With the help of MathOverflow, at https://mathoverflow.net/questions/360363/, I can now turn zhoraster's suggestion into a complete answer:

Define the stochastic process $(\tilde{W}_t^{(\lambda)})_{t \geq 0}$ such that $\tilde{W}_t^{(\lambda)}$ agrees with $W_t^{(\lambda)}$ at every second event of the Poisson process $(N_{\lambda t})_{t \geq 0}$ and is linearly interpolated in between. That is to say, $$ \tilde{W}_{\frac{1}{\lambda}(rS_{2n} \, + \, (1-r)S_{2n+2})}^{(\lambda)} \ = \ rW_{\frac{1}{\lambda} S_{2n}}^{(\lambda)} \ + \ (1-r)W_{\frac{1}{\lambda} S_{2n+2}}^{(\lambda)} \quad \textrm{for all } n \geq 0, \, r \in [0,1] $$ where $$ S_n \, := \, \inf\{t \geq 0 : N_t = n\}. $$ Let $D_n=S_{n+1}-S_n$. We can apply the generalised Donsker invariance principle in the linked MO question, with $\Delta_n=D_{2n}+D_{2n+1}$ and $X_n=D_{2n}-D_{2n+1}$, to yield that on any compact time-interval the process $\tilde{W}_t^{(\lambda)}$ converges in distribution (w.r.t. the topology of uniform convergence) to the Wiener process.

So it only remains to control the difference $\tilde{W}_t^{(\lambda)} - W_t^{(\lambda)}$. For any $T>0$, $$ \max_{t \in [0,T]} |\tilde{W}_t^{(\lambda)} - W_t^{(\lambda)}| \ \leq \ \frac{\max\{ D_n : 0 \leq n \leq N_{\lambda T}+1 \}}{\sqrt{\lambda}}. $$ Now for any $\varepsilon>0$, taking sufficiently large $\lambda$ will give that $\mathbb{P}(N_{\lambda T}+1 \leq \lceil 2\lambda T \rceil)>1-\frac{\varepsilon}{2}$; and since $$ \max\{ D_n : 0 \leq n \leq \lceil 2\lambda T \rceil \} - \log(\lceil 2\lambda T \rceil) $$ is convergent in distribution as $\lambda \to \infty$ (to the Gumbel distribution) and $\frac{\log(x)}{\sqrt{x}} \to 0$ as $x \to \infty$, it follows that for sufficiently large $\lambda$, $$ \mathbb{P}\left( \frac{\max\{ D_n : 0 \leq n \leq \lceil 2\lambda T \rceil \}}{\sqrt{\lambda}} > \varepsilon \right) \ < \ \tfrac{\varepsilon}{2} $$ and therefore $$ \mathbb{P}\left( \frac{\max\{ D_n : 0 \leq n \leq N_{\lambda T}+1 \}}{\sqrt{\lambda}} > \varepsilon \right) \ < \ \varepsilon. $$ Hence $\max_{t \in [0,T]} |\tilde{W}_t^{(\lambda)} - W_t^{(\lambda)}|$ converges in probability to $0$, and so it follows that $(W_t^{(\lambda)})_{t \in [0,T]}$ converges in distribution to the Wiener process on $[0,T]$.