Scaling idea

Let

$$ X_n=\sum_{i=1}^n \varepsilon_i $$

be a symmetric random walk with $\varepsilon_i=\pm 1$ equally likely.

For fixed $t>0$, consider the rescaled position

$$ \frac{1}{\sqrt n}X_{\lfloor nt\rfloor}. $$

Limiting distribution

By the central limit theorem,

$$ \frac{1}{\sqrt n}X_{\lfloor nt\rfloor} \xrightarrow{d} N(0,t). $$

The variance is

$$ \mathrm{Var}\left(\frac{1}{\sqrt n}X_{\lfloor nt\rfloor}\right)

\frac{\lfloor nt\rfloor}{n} \to t. $$

Why this matters

Brownian motion is the continuous-time, continuous-space limit of symmetric random walk under this joint time-space scaling.