Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2012 / S9 / 2 / 1

Independent identically distributed increments have the same marginal distribution at every step and factorizing joint distributions across steps. With finite mean $\mu$ and variance $s^2$, an additive random walk has mean displacement $n\mu$ and variance $n{s}^2$ after $n$ steps. Zero increment autocovariances follow from independence, but their converse is false. Gaussian increments are an additional assumption, not part of the definition of a random walk.

To investigate a measured price process, first specify the sampling interval and whether the model concerns price differences or log returns. Remove an estimated drift and examine stationarity, increment distributions, autocorrelations and variance growth over nonoverlapping horizons. Also test higher-order statistics, absolute or squared-return correlations, conditional distributions and out-of-sample predictability; those can reveal dependence missed by linear autocorrelation. Check whether different time scales and market regimes give compatible results, and account for bid-ask effects and intraday seasonality at short lags. Finite data cannot prove independence or permanent stationarity. Failure to reject a particular test is weaker than establishing that every joint distribution factorizes; the next example makes that distinction explicit.