A random walk without drift is not stationary. Because its autocovariance function depends on time. A random walk with drift is not stationary as its mean is not constant. But what is the autocovariance of a random walk with drift? Is it the same as that of a random walk with no drift?
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You can obtain the expression of the autocovariance function. Start writing the equation of each process recursively: for the random walk: $y_t=\underbrace{y_{t-1}}_{y_{t-2}+\varepsilon_{t-1}}+\varepsilon_t\to$ $y_t=y_0+\sum_{i=1}^t\varepsilon_i$; for the r.w. with drift: $z_t=\underbrace{\alpha+z_{t-1}}_{\alpha+z_{t-2}+\varepsilon_{t-1}}+\varepsilon_t\to$ $z_t=z_0 + \alpha t + \sum_{i=1}^t\varepsilon_i$. You can assume that the process starts at a fixed value, e.g., $y_0=z_0=0$. Upon these expressions you can obtain the mean and the autocovariances. – javlacalle Oct 20 '18 at 12:57
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Related posts: [this](https://stats.stackexchange.com/questions/181167/) and [this](https://stats.stackexchange.com/questions/140836/). – javlacalle Oct 20 '18 at 13:01