White Noise, Random Walks, and Stationary Autocorrelation
Summary
The document distinguishes white noise from a random walk using their defining properties. White noise is described as a zero-mean process with constant variance and no autocorrelation; a random walk accumulates white-noise increments. Consequently, the walk’s variance grows over time, so it is neither white noise nor stationary under this definition. Differencing a random walk recovers its increments, linking the two processes in time-series analysis.
It also separates weak stationarity from independence: white noise is stationary up to second order, while an independent and identically distributed noise sequence is strongly stationary. A stationary process can still have autocorrelation, illustrated by an AR(1) process whose autocorrelation decays geometrically when the autoregressive coefficient has magnitude below one. These are conceptual examples rather than empirical results; the definitions matter, and the random-walk variance argument presumes nonzero innovation variance.
Key ideas
- A random walk is formed by cumulatively adding white-noise innovations.
- The variance of a random walk with nonzero innovation variance increases over time.
- Differencing a random walk yields its innovation sequence.
- White noise is second-order stationary under the stated definition, while independent identically distributed noise is strongly stationary.
- A stationary AR(1) process can have nonzero autocorrelation.
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Full text
# Relationships between white noise and random walk
# 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?
Thank you in advance.
## Answer by AFK (score 12, accepted)
https://quant.stackexchange.com/a/19456
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 $|\varphi| < 1$.
## Answer by Alex C (score 5)
https://quant.stackexchange.com/a/19463
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]Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.