# Sufficient conditions for a Brownian motion

1. Sep 16, 2014

### economicsnerd

I'm given a probability measure $\mathbb P$ on $\Omega = \{f\in C([0,1],\mathbb R): \enspace f(0)=0\}$ and told that $\mathbb P$ satisfies i.i.d. increments.

I'm interested in the weakest additional conditions that will ensure that $\mathbb P$ describes a Brownian motion, i.e. that there is some $\mu\in\mathbb R$ and $\sigma\in\mathbb R_+$ such that $\mathbb P$ is the law of $X$ as described by $dX_t = \mu dt + \sigma dZ_t$ for a standard Brownian motion $Z_t$.

Does it already follow from the assumptions of i.i.d. increments and continuity? It seems like I should be fine (by CLT) as long as increments have well-defined mean and finite variance. Do these properties come for free? If not, can I get away with assuming less? For instance, is it enough to only assume finite variance?