Which statement regarding random walk series is true?

In the context of random walk series, the assertion that the variance increases as time increases is a fundamental characteristic of this type of stochastic process. A random walk is defined as a sequence of steps, where each step is determined by a random variable. As more steps are taken, the potential distance from the starting point increases, leading to an increase in the variance.

Specifically, in a simple random walk, if each step is independent and identically distributed with a finite variance, the variance after 'n' steps will increase proportionally to 'n'. This reflects the accumulation of variability over time, as each additional step contributes to the overall uncertainty in the position of the series.

In contrast, the other options present characteristics that do not align with the properties of a random walk. For instance, the variance does not decrease over time, nor does it remain constant; a stationary series would imply that statistical properties like mean and variance do not change over time, which is not the case for a random walk. Therefore, the correct statement regarding the behavior of variance in a random walk series is that it indeed increases as time progresses.