Which of the following statements about white noise processes is false?

The statement that all white noise processes are stationary is indeed correct. A white noise process is characterized by having a constant mean, constant variance, and no autocorrelation at any lag other than zero, which fulfills the criteria for stationarity.

Focusing on the other statements:

The variance of a random walk does not remain constant over time; instead, it increases as more observations are added to it. Specifically, for a random walk, the variance of the sum of the random variables increases with time, indicating that the process isn't stationary.

First-order differencing is a technique applied to achieve stationarity in a non-stationary series, such as a random walk. However, since a random walk itself is non-stationary, after applying first-order differencing, one may introduce dependency among the observations that did not exist in the original random walk.

Lastly, a key feature of a random walk is that its variance grows over time, which reflects the accumulating uncertainty about future values as the process evolves.

Thus, the correct statement about white noise processes being stationary clearly aligns with the definition and properties of white noise, distinguishing it from the other misleading statements.