= Solution
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> $ns^2$ 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.
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