# Random walk scale invariance?

1. Oct 11, 2012

### sjweinberg

Consider a random walk (in any dimension) with $N$ steps and a step size of 1. Take a real number $\alpha > 0$ and consider another random walk which takes $\alpha^2 N$ steps but wil step size $\frac{1}{\alpha}$.

I immediately noticed that the mean deviation after the full walk in both cases is the same: $\frac{1}{\alpha} \sqrt{\alpha^2 N} = \sqrt{N}$. However, I'm curious to what extent these two random walks look identical. If were to take a Brownian motion type limit (where N becomes large and the step size 1 is thought of as being small), would the walks look identical?

Thanks in advance to any masters of statistics.

2. Oct 11, 2012

### chiro

3. Oct 11, 2012