A random walk process (no drift!) has mean zero, that's why it's weakly stationary but not strong (strict) stationary

Question

BlackTom AI · Accepted Answer

Consider the statement: 'A random walk process (no drift!) has mean zero, that's why it's weakly stationary but not strong (strict) stationary'.

Option 1: True. The claim that the process has mean zero is accurate for a zero-drift random walk, but stationary properties depend on more than just the mean. Weak (second-order) stationarity requires constant mean and constant autocovariances that depend only on lag. In a random walk, while the mean can be zero (for a zero-drift walk), the variance grows with time, so the process does not have a constant variance. Therefore it is not weakly stationary. This already contradicts the statement asserting weak stationarity, so this option mischaracterizes the process.

Option 2: False. The core issue is that a random walk with no drift does not have a constant variance; Var(S_t) = t σ² if the step variances are σ², which increases with t. Since weak stationarity requires constant mean and constant autocovariances (which would imply a constant variance), the changing variance means the process is not weakly stationary. Strong (strict) stationarity would require that the finite-dimensional distributions are invariant under time shifts, which a random walk does not satisfy because the distribution of S_t changes with t (variance grows), so it is not strictly stationary either. The claim that the process is weakly stationary due to mean zero is therefore incorrect.

Additional nuance: Some texts distinguish between a random walk with drift and without drift; in either case, the key property for weak stationarity fails due to the time-varying variance. The presence of drift affects the mean over time, but even without drift, the non-constant variance breaks weak stationarity, and strict stationarity would still fail because the distribution changes with time. The statement conflates the mean condition with the full set of requirements for stationarity.

In sum, the reasoning shows that the assertion is incorrect because the necessary condition for weak stationarity—constant mean and constant variance (and, more generally, constant autocovariances)—is violated by a zero-drift random walk due to its variance increasing with time, and thus the claim cannot be true.

A random walk process (no drift!) has mean zero, that's why it's weakly stationary but not strong (strict) stationaryTrue/False

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