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A common characaterization of Wiener's process is the following which I took directly from Wikipedia:

  1. $W_0 = 0$ a.s.
  2. $W$ has independent increments: $W_{t+u} - W_t$ is independent of $σ(W_s : s ≤ t)$ for $u ≥ 0$
  3. $W$ has Gaussian increments: $W_{t+u} - W_t$ is normally distributed with mean $0$ and variance $u$, $W_{t+u}−W_{t} \sim N(0, u)$
  4. $W$ has continuous paths: With probability 1, $W_t$ is continuous in t.

I think to remember that I once heard you can replace the normality assumption (3) by something weaker. Is this true? Maybe by the right first and second moments?

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That's interesting. Seems to me the fact that you can replace (3) by something weaker follows from the Central Limit Theorem, or at least by a CLT-ish argument. I wouldn't know what the fanciest result in this direction would be - there are all sorts of versions of things like the CLT, but it seems clear that at the very least one could replace (3) by the assumption that there exists $X$ with mean $0$ and variance $1$ such that $$W_{t+u}-W_t\sim X/u^{1/2}.$$Because then, noting that $$W_{t+u}-W_t= \sum_{j=1}^n(W_{t+j/n}-W_{t+(j-1)/n}),$$the independence of the increments shows via CLT that $W_{t+u}-W_t$ is arbitrarily close to normal, hence is normal.

That's using the absolute plain vanilla CLT, for sums of iid variables with finite variance. Fancier versions of CLT will allow versions of (3) with weaker hypotheses.

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