Why differentiating a non-stationary time series can lead to stationarity?

Question

What is the mathematics rationale behind it? I can get somehow the intuition by looking at plots of the differentiated series. For instance, the trend on the stock prices time series is removed by differentiating it. But I would like to see a more formal proof on why it works.

Answer 1

It can get rid of dependence structures.

For example, consider the process as follows: $X_{t} = t + w_{t}$ where $w_{t}$ is a white noise process.

$E[X_{t}]=t$ is not a constant (it depends on $t$).

$E[X_{t}-X_{t-1}]=E[t-t+1+w_{t}-w_{t-1}]=1$

$\gamma(h) = E[X_{t}-X_{t-1},X_{t+h}-X_{t+h-1}]=E[(t-t+1+w_{t}-w_{t-1})(t+h-t-h+1+w_{t}-w_{t-1})]-1 =E[(1+w_{t}-w_{t-1})(1+w_{t+h}-w_{t+h-1})]-1$

is independent of $t$.

Thus the differenced series is second order stationary.