Simple application of Donsker's theorem

Question

I am trying to do exercise 5.15 in Moerter's book "Brownian Motion". It seems quite easy, but I can't solve it somehow: Suppose $S(j)_j$ is a SRW on the integers, started at zero. Show that: $$ \frac{1}{n^2}\min_j\{|S(j)|=n\}\to^d\min_t\{|B(t)|=1\}\, . $$ Now, this looks like a standard application of Donsker's theorem (although, perhaps it is not?). Anyway, I tried hitting it with Donskers theorem, but I just keep getting nonesense: $$ P(\min_t\{|B(t)|=1\}\le s)=P(M_s\ge 1)=P(B\in K_s)\, \ $$ where $M$ is the max of the modulus of a Brownian motion and and $K_s$ is the set of continuous functions whos maximum is bigger than one. But: $$ P(S^*(t)\in K_s)=? $$ The first problem is that we don't have the intervall $[0,1]$ as required in Donsker's theorem, and the second one is that I cannot related the interpolated and scaled SRW to the property above (the one with $1/n^2$) Any help welcomed. thank you!

Answer 1

Let $(X_i)_{i\in\mathbb{N}^*}$ be iid. random variables of law $\mathbb{P}(X_1 = \pm 1) = 1/2$. Let $S_0 = 0$ and $S_n = X_1+\ldots+X_n$ for all $n \in\mathbb{N}^*$. Then, let $S_t = S_{\lfloor t \rfloor}+\{t\} X_{\lfloor t \rfloor +1}$ for all $t\geq0$. If $ S_t^n = \frac{S_{nt}}{\sqrt{n}}$ for all $n\in\mathbb{N}^*,t\geq 0$, then Donsker's theorem states that $S^n$ converges in the polish space $C(\mathbb{R}_+,\mathbb{R})$ to a brownian motion $B = (B_t)_{t\geq0}$.

First, notice that $S_t$ is an integer iff $t$ is a integer (this is true because of the choice of the law of $X_1$) so we have, for all $n\geq 1$, $$ \inf\{j\in\mathbb{N}^*: |S_j| = n\} = \inf\{t\geq0: |S_t| = n\}.$$ Then notice how $$ |S_t|= n \iff \frac{|S_{n^2 t/n^2}|}{n}=1\iff |S_{t/n^2}^{n^2}| = 1. $$ This leads to $$\inf\{j\in\mathbb{N}^*: |S_j| = n\} = \inf\{t\geq 0: |S_{t/n^2}^{n^2}| = 1\} = n^2\inf\{t\geq 0: |S_{t}^{n^2}| = 1\}. $$ Lastly, since $\Phi : f \in C(\mathbb{R}_+,\mathbb{R}) \mapsto \inf\{t\geq 0: |f(t)| =1\}$ is continuous almost surely at $B$ (this non-trivial fact follows from the fact that a brownian trajectory has almost surely no local extrema), we know $$\frac{1}{n^2}\inf\{j\in\mathbb{N}^*: |S_j| = n\} = \Phi(S^{n^2}) $$ converges in law to $\Phi(B) = \inf\{t\geq0: |B_t|=1\}. $

We lastly used this little lemma: if $X_n$ converges in law to $X$ and $f$ is almost surely continuous at $X$, then $f(X_n)$ converges in law to $f(X)$. To prove this, one can use Skorokhod's theorem.