Consider a set of time series data defined as "random walk" as follows:

$x_t=+x_{t-1}+w_t$

for $t=1,2,$dots with $x_0=0,$ and where $w_t$ is white noise with variance ${}_w^2$

Define the "first$-$difference" time series data by:

$x_t=x_t-x_{t-1}$

Prove mathematically $($by derivation $-$ NOT computationally, such as Python modeling$)$:

$($a$)$ the random walk process is not stationary

Hint: one way to demonstrate this claim is to show that the "random walk" data can be

written as:

$x_t=t+{\displaystyle \underset{k=1}{\overset{t}{}}}{w}_k$

$($b$)$ the "first difference" transformation yields a weakly stationary process