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Is it true that the mean of a random walk process does not depend on time and the sequence can be considered mean stationary?

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TLDR: Yes

Let us consider a random walk in discrete time: At every timestep $t = n\Delta t$ with $n \in \mathbb{N}$ the process goes one step forward or one step back with equal probability.
Leading:
$P(\Delta x_i = 1) = 1/2$
$P(\Delta x_i = -1) = 1/2$
(where $i$ denotes the timesteps)

The process is at $t = n\Delta t$ is $X_t = \sum_{i=0}^{n}{\Delta x_i}$.
Taking the average $\mathbb{E}X_t = \mathbb{E}(\sum_{i=0}^{n}{\Delta x_i}) = \sum_{i=0}^{n}{\mathbb{E}(\Delta x_i)} = \sum_{i=0}^{n}{(\frac{1}{2}(1)+\frac{1}{2}(-1))} = 0$. This is clearly not dependent on time.

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