# Testing whether a process is Wiener process

Ideally I would like links to code implementations (eg. Matlab ) or book references, but I would appreciate suggestions on various methods.

We start with sampled process $X_{t}$.

1. A straightforward way is through repeated normality tests:

a. Test for normality of $X_{t}$ using KS test

b. Independence of increments: we test whether the product $X_{T/2}(X_{T}-X_{T/2})/\sigma^{2}$ is close to a standard normal, where $\sigma^{2}=(T/2)^{2}$ .

c. Joint normal for increments $Y_{n}=(X_{t_2}-X_{t_1},...,X_{t_n}-X_{t_n-1})$: we divide the possible values of $Y_{n}$ into $m$ cells and denote $O_{j}:=$# samples that fall into cell $j$. Then the statistic $\sum \frac{(O_{j}-E[O_{j}])^{2}}{E[O_{j}]}$ should be approximately a $\chi^{2}-$distribution.

2. Maybe testing for quadratic variation=$t$ and martingale property (Levy characterization)?

3. Some spectral characterization for WP?

4. The other link is testing for fractional WP, which is more involved.

5. Turning it into a test for White noise since it is the "derivative" of WP. So maybe taking finite difference for WP and doing white noises tests for it: $$(X_{t+\Delta t}-X_{t})/\Delta t.$$