Exercise: Why $\sqrt{n}$ Is the Right Rescaling for Donsker

Prerequisites: Random Walk, Central Limit Theorem

Problem

Let $S_n = \sum_{i=1}^{n} X_i$ be a simple symmetric random walk with $X_i = \pm 1$ i.i.d.

1. Consider the rescaling $W_n^{(\alpha)}(t) := n^{-\alpha} S_{\lfloor nt \rfloor}$ for $t \in [0, 1]$. Compute $\operatorname{Var}(W_n^{(\alpha)}(t))$ in the limit $n \to \infty$ for each of the three cases: $\alpha < 1/2$, $\alpha = 1/2$, and $\alpha > 1/2$.
2. Explain, using your answer to part 1, why each of the non-$\frac{1}{2}$ choices fails to produce a meaningful limit process.
3. Under the $\alpha = 1/2$ rescaling, what is $\operatorname{Cov}(W_n^{(1/2)}(s), W_n^{(1/2)}(t))$ for $0 \le s \le t \le 1$ as $n \to \infty$? Compare to the known covariance structure of Brownian motion.
4. (Conceptual) In one sentence, explain why the dimensional argument "time scales as $n$, space scales as $\sqrt{n}$" follows directly from the variance formula $\operatorname{Var}(S_n) = n$ and nothing else about the underlying distribution.

Hint

For part 1, $\operatorname{Var}(S_n) = n$ so $\operatorname{Var}(n^{-\alpha} S_n) = n^{1-2\alpha}$. For part 3, use $\operatorname{Cov}(S_m, S_n) = \min(m, n)$, which follows from the independent-increments property.