Random walk without drift is a stationary process

Question

BlackTom AI · Accepted Answer

The statement to evaluate is: 'Random walk without drift is a stationary process.'

First, consider what a random walk without drift means. In a classic simple random walk, the position at time t is the sum of independent random shocks with zero mean (drift = 0). Each step adds a new random value, so the displacement after t steps is the cumulative sum of those shocks.

Now, recall the definition of a stationary process. A process is stationary if its statistical properties do not change over time. For strict stationarity, the joint distributions are invariant to time shifts; for wide-sense (second-order) stationarity, the mean and autocovariance do not depend on time.

In a random walk without drift, the mean of the position at time t is zero if the shocks have zero mean, but the variance grows with time: Var(X_t) = t * Var(step). This increasing variance with t implies the process does not have constant variance over time, so it is not wide-sense stationary.

Additionally, because the variance grows with time, the autocovariance structure depends on t, t−s, and thus changes over time, which violates stationarity (even in the wide-sense sense).

If a random walk had nonzero drift, the mean would grow linearly with time, which still does not yield stationarity for the same reason—the variance term dominates the behavior and grows with time, so drift does not restore stationarity.

Another way to see this is to note that a stationary process has a stable distribution over time; a random walk accumulates shocks, so its distribution spreads out without bound as time increases, incompatible with stationarity.

Therefore, the proposition that a random walk without drift is stationary contradicts the core property of variance growing with time, making the statement false.

In summary, because the variance of a driftless random walk increases linearly with time, the process is non-stationary, and the claim provided is incorrect.

Random walk without drift is a stationary process判断题

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