How to prove nonstationarity of a random walk? Given the AR(1) model $Y_t = ϕY_{t−1} + e_t$ . I want to show if |ϕ| = 1, the process cannot be stationary. I know to prove stationary it suffices to prove either mean function or autocovariance function is not independent of time. It is my first time approaching this type of problem so I really have no clue how to approach it.
 A: As you said, it suffices to show that the Variance (the 0th-order of the autocovariance) is time dependent.
$y_{t}=y_{t-1}+\varepsilon_{t}$
$Var\left(y_{t}\right)=Var\left(y_{t-1}+\varepsilon_{t}\right)$
$Var\left(y_{t}\right)=Var\left(y_{t-1}\right)+\sigma_{\varepsilon}^{2}$
$Var\left(y_{t}\right)=Var\left(y_{t-2}+\varepsilon_{t-1}\right)+\sigma_{\varepsilon}^{2}$
$Var\left(y_{t}\right)=Var\left(y_{t-2}\right)+2\sigma_{\varepsilon}^{2}$
repeating this for $t$ steps:
$Var\left(y_{t}\right)=Var\left(y_{0}\right)+t\sigma_{\varepsilon}^{2}$
If we assume that $y_{0}$ is given:
$Var\left(y_{t}\right)=t\sigma_{\varepsilon}^{2}$
Which means that the variance of the process increases with time, and therefore not stationary.
Hope this is what you are looking for.
A: First, I think you mean:

I know to [dis]prove stationar[it]y it suffices to prove [that] either [the] mean
  function or [the] autocovariance function is not independent of time.

But yeah that's right. So here's a hint. I would focus on the autocovariance portion. If this process is indeed stationary, then the (auto?)covariance function 
$$
\gamma_X(t+h,h) \overset{\text{def}}{=} \operatorname{Cov}(X_{t+h},X_t)
$$
shouldn't depend on time ($t$) for any lag $h$. So to disprove that, you would have to find some particular $h$ where it does depend on time.  
