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

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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.

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When you say correlation between values at different times... do you think all possible lag combinations or only t vs t-1? – user333 Feb 10 '11 at 22:37
@user333 All nonzero lags: that's the first equation in the Wikipedia link @onestop gave. – whuber♦ Feb 10 '11 at 22:40
you forgot the constant variation. If variation varies, then the process is not white noise. – mpiktas Feb 11 '11 at 7:37
@mpiktas: you're right, good point. – onestop Feb 11 '11 at 8:51
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@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" :) – Dmitrij Celov Feb 14 '11 at 10:34
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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.

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