# Wiener Process and Random Walk

Quoted from Wikepedia:

The Wiener process can be constructed as the scaling limit of a random walk, or other discrete-time stochastic processes with stationary independent increments. This is known as Donsker's theorem.

I was wondering if the scaling limit of any random walk is always a Wiener process, or just the scaling limits of some special kinds of random walks are, such as Gaussian random walk?

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In light of your other question, one should say there are a lot of things referred to by the name "random walk". In its simplest form, Donsker's theorem is about a process $X_n$ in $\mathbb{R}^d$ whose increments $X_n - X_{n-1}$ are iid with any distribution that has zero mean and finite variance. In particular, the increments do not have to be Gaussian. It then asserts that the scaling limit of the process $X_n$ is $d$-dimensional Brownian motion (aka Wiener process).