I would like to ask 5 questions about relations between these processes.
1) Could white noise be also a random walk?
2) Could random walk be also a white noise?
3) Could white noise be stationary?
4) Could random walk be stationary?
5) Could stationary process have autocorrelation?
Thank you in advance.
I will assume a white noise is a process $(\varepsilon_t)$ with zero mean, no autocorrelation and constant variance $\sigma^2 > 0$ while a random walk is a process $(x_t)$ defined by $$ x_{t+1} = x_t + \varepsilon_{t+1} $$ where $\varepsilon$ is a white noise.
1) No since $Var(x_{t+1}) = Var(x_t) + Var(\varepsilon_{t+1})$ is stricly increasing while the variance of a white noise is constant.
2) No same reason as above.
3) By definition, it is stationary up to order 2. A strong white noise (i.e. an i.i.d sequence) is strongly stationary.
4) No, again because $Var(x_{t+1}) > Var(x_t)$.
5) Yes, the simplest example is an AR(1) process $$ x_{t+1} = c + \varphi x_t + \varepsilon_{t+1} $$ It has autocorrelation $\rho(j) = \varphi^j$ and it is stationary if $|\varphi| < 1$.
Regarding the relationship between white noise and a random walk, I would put it this way: a random walk is integrated white noise. [And vice versa we get a white noise when we differentiate/difference a random walk]. Or to put it in quant finance terms: white noise is like the daily changes in the S&P in points, a random walk is the S&P daily level itself.
So, just for fun, of these two time series, which is the white noise and which is the random walk?
[2115,2120.5,2117.1,2097.4,2113.4,2114.2,2098.5,2101.6,2099.1,2108.3,2091.3]
[5.5,-3.4,-19.7,16,0.8,-15.7,3.1,-2.5,9.2,-17]
