I am a bit confused about how to fix some feedback I got.
For a paper I am writing for a class, I used the following to explain why a certain random walk converges to Brownian motion, the following is a direct snip from my paper:
Definition (Convergence in distribution)
A sequence of random variables $X_1,X_2,\dots,$ converges in distribution to a random variable $X$ if
\begin{equation}
\lim _{n \rightarrow \infty} F_{X_{n}}(x)=F_{X}(x)
\end{equation}
at all points $x$ where $F_{X}(x)$ is continuous. Where $F_{X_{n}}$ and $F_{X}$ are the cumulative distribution functions of random variables $X_n$ and $X$ respectively.
This scaled random walk on $[0,1]$ then converges to a one-dimensional standard Brownian motion by the following theorem:
Theorem (Donsker's Theorem )
Let $\{X_i , i \geq 1 \}$ be a sequence of real-valued i.i.d. random variables such that $\mathbb{E}(X_1) = 0$, and $\operatorname{Var}(X_1) = 1$. Let $\{ S_n , n \geq 0 \}$ be the one-dimensional random walk based on $\{X_i , i \geq 1 \}$ and define $\left\{W_{n}(t), 0 \leq t \leq 1\right\}$ as $W_n(0) = 0$ and
\begin{equation}
W_{n}(t)=\frac{1}{\sqrt{n}} S_{\lfloor n t \rfloor }+(n t-\lfloor n t \rfloor) \frac{1}{\sqrt{n}} X_{\lfloor
n t \rfloor
+1}, \quad t \in(0,1], n \geq 1 \label{Donsker}
\end{equation}
Then, $W_{n}$ converges in distribution to $\mathbb{W}$ as $n \rightarrow \infty$, where $\mathbb{W}$ denotes the one-dimensional standard Brownian motion.
The feedback I received said that I cannot directly say $W_{n}$ converges in distribution to $\mathbb{W}$, as $W_{n}$ is not one random variable but a random function. I have tried to find a similar definition for random functions however I havent had much luck. Does anyone know a reasonable way to fix this?
Best Answer
Here convergence of $W_n$ to $W$ in distribution means convergence of the induced measures on the space $C[0,1]$. $W_n$ induces a measure $\mu_n$ on this space and $W$ induces the Wiener measure $\mu$. $\mu_n \to \mu$ means $\int Fd\mu_n \to \int F d\mu$ for every bounded continuous function $F:C[0,1] \to \mathbb R$.
We refer to $\mu_n$ and $\mu$ as the distributions of the processes $W_n$ and $W$.