# Relationships between white noise and random walk

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?

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 $|\varphi| < 1$.

• As per your 3rd point, does it mean that white noise is strictly stationary ? – Neeraj Aug 26 '15 at 6:19

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]