Which statement about the characteristics of a random walk is true?

A random walk is a statistical process in which the current value is composed of the previous value plus a random disturbance, making it non-stationary. In such a model, the characteristic that its variance increases over time is crucial.

As a random walk progresses, the cumulative effects of the random disturbances compound, which leads to an expanding range of possible values. More specifically, if we consider the changes over time, the variance of the process actually grows linearly with time. For example, if we denote the variance of the random walk at time ( t ) as ( Var[X_t] ), it will increase as ( Var[X_t] = t \cdot Var[\epsilon] ), where ( \epsilon ) denotes the random disturbances. Hence, the statement indicating that the variance increases over time accurately reflects a fundamental property of random walks.

On the other hand, other statements fail to hold true for a random walk. The idea that it is stationary contradicts the definition since a stationary process shows constant mean and variance over time. Similarly, stating that the mean remains constant misrepresents the random walk's tendency to drift away from any fixed mean, and the assertion of constant variance does not apply due to the increasing nature of