# What is a white noise process?

What is the best way of defining white noise process so it is intuitive and easy to understand?

• Noises (stationary random series) are classified by a PSD color indicating how energy is distributed over frequency . White has the meaning of uniform distribution (flat), like in white light or white acoustic noise. White noise is made of all frequencies in equal proportions (not found in nature). Red: Energy decreases with frequency. Pink, Between white and red: Energy decreases, but is equal per octave, common in nature. Blue: Energy increases (radioactivity). See Colors of noise
– mins
Commented Aug 8, 2023 at 12:31

A white noise process is one with a mean zero and no correlation between its values at different times. See the 'white random process' section of Wikipedia's article on white noise.

• When you say correlation between values at different times... do you think all possible lag combinations or only t vs t-1? Commented Feb 10, 2011 at 22:37
• @user333 All nonzero lags: that's the first equation in the Wikipedia link @onestop gave.
– whuber
Commented Feb 10, 2011 at 22:40
• you forgot the constant variation. If variation varies, then the process is not white noise. Commented Feb 11, 2011 at 7:37
• @mpiktas: you're right, good point. Commented Feb 11, 2011 at 8:51
• @mpiktas, i usually explain the white noise to students through the spectral density concept, at least it gives light to (through the analogy with white color) why the noise is "white", and why the $AR(1)$ process could be could called "red" and there is no "black noise" :) Commented Feb 14, 2011 at 10:34

A white noise process is a random process of random variables that are uncorrelated, have mean zero, and a finite variance. Formally, $X(t)$ is a white noise process if $$E(X(t)) = 0, E(X(t)^2) = S^2\text{, and } E(X(t)X(h)) = 0 \text{ for } t\neq h\text{.}$$ A slightly stronger condition is that they are independent from one another; this is an "independent white noise process."

I myself usually think of white noise as an iid sequence with zero mean. At different times values of the process are then independent of each other, which is much stronger requirement than correlation zero. What is the best with this definition that it works in any context.

Side note. I only explained my intuition, the correct definition of white noise is given by @onestop. The definition I gave is mathematically defined as white noise in strict sense.