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"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.

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    $\begingroup$ What is $W^{(n)}(t)\,?$ BTW: I am quite certain that Steven Shreve never wrote a book called "continuos models". $\endgroup$
    – Kurt G.
    Commented Aug 1, 2023 at 8:46
  • $\begingroup$ @KurtG. I think the OP is somehow trying to refer to the book "Stochastic Calculus for Finance II Continuous-Time Models". $\endgroup$
    – Alper
    Commented Aug 1, 2023 at 12:19
  • $\begingroup$ 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. $\endgroup$
    – Quasar
    Commented Aug 1, 2023 at 17:45

2 Answers 2

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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 $W^n(t)$.

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  • $\begingroup$ I remember Hamilton's "time series analysis" having a nice proof. $\endgroup$
    – mark leeds
    Commented Sep 4, 2023 at 15:32
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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 $W^{(n)}(t)$.

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