How to apply CLT on scaled symmetric random walk--Shreve unclear

"Theorem 3.2.1 (Central limit)" in the book "Stochastic Calculus for Finance II Continuous-Time Models" by Steven Shreve says:

Theorem. Fix $$t\geq0$$. As $$n\to \infty$$, the distribution of the scaled random walk $$W^{(n)}(t)$$ evaluated at time $$t$$ converges to the normal distribution with mean zero and variance $$t$$.

A "proof" is given there, without using the general Central Limit Theorem.

However, my question is the following: Suppose we wish to simply apply the CLT, how can we even do that here?

For a fixed $$t$$, the sequence $$W^{(n)}(t) \quad (n= 1,2,3, \ldots)$$ is not i.i.d. (in fact, they are of course identically distributed, but they are not independent!). Therefore I can't see how we can deduce this using the CLT.

• What is $W^{(n)}(t)\,?$ BTW: I am quite certain that Steven Shreve never wrote a book called "continuos models". Commented Aug 1, 2023 at 8:46
• @KurtG. I think the OP is somehow trying to refer to the book "Stochastic Calculus for Finance II Continuous-Time Models". Commented Aug 1, 2023 at 12:19
• In Shreve's terminology, $W^{(n)}(t) = \frac{1}{\sqrt{n}}M_{nt}$, where $(M_t:t=0,1,2,\ldots)$ is symmetric random walk with step-size $1$. There are several CLT's with different conditions that need to hold, in order to apply them. By the way, out here, $W^{(n)}(t) - W^{(n)}(s)$ and $W^{(n)}(t + a) - W^{(n)}(s + a)$ are IID. Commented Aug 1, 2023 at 17:45

Using the phrasing of the central limit from Wolfram, you can think of it as the central limit theorem being applied to the (scaled) $$X_i$$ "coin tosses" that make up $$W^n(t)$$.
If I recall correctly, Shreve defines $$W^{(n)}(t)$$ as constructed from increments: $$W^{(n)}(t):=\sum_{i=1}^{tn}\frac{1}{\sqrt{n}} S_{i}$$ where $$S_{i}$$ are symmetric Bernoulli random variables.
When expressed as a scaled sum of i.i.d. increments, the CLT applies and we can deduce the distribution of Brownian Motion as the limit of $$W^{(n)}(t)$$.