I'm a math graduate student and I have to use time series in my thesis. I have not so much knowledge in statistics, but I've studied about probability and time series. So my question maybe can be very simple for an statistician and it is: why we can suppose the white noises normally distributed? In general, when we can suppose normal distribution about a random variable? Is there any approximation theory in statistics that support this assumption (CLT?)? Thanks!
Is there any approximation theory in statistics that support this assumption (CLT?)?
As already noted, white noise does not need to be normally distributed and I would assume that you find deviations from normality in any noise if you only look hard enough (i.e., take sufficiently many samples).
However, in reality a random variable can often be regarded as the sum of many smaller random variables. For example, if we measure the speed of birds flying past a window (and consider it as a random variable), we can break it down into the following:
- the physical fitness of the individual bird.
- the daily form of the bird
- the current wind speed
- the presence of incentives for the bird to acquire a certain speed (e.g., potential prey or predators)
- the measurement noise of whatever device we use to measure the bird’s speed
Now, the distributions of some of these influences may not be normal, e.g., if we have two dominant bird species of different sizes, the first one will likely be bimodal. But if we have sufficiently many such influences contributing to our measurement, the Central Limit Theorem yields that the distribution of our summed observable tends to be close to normality.
Due to this, we encounter many approximately normally distributed variables in application and thus normality is a good default assumption. Moreover, in many cases where we strongly deviate from normality (e.g., the bimodal bird fitness distribution from above), it’s no intellectual challenge to predict this – which further reduces the risk of errors due to falsely assuming normality.
Sidenote: This is probably the most important consequence of the Central Limit Theorem, which is sadly rarely mentioned in education. For example, standard error propagation relies on the assumption of normality and thus on the Central Limit Theorem.
There is no general, mathematical, and universal reason for assuming white noises to be normally distributed. They can be of many other distributions, too. For instance, the daily income cannot have white noise since it cannot be negative.
On the other hand, there are many useful tools where the assumption of normality is in place. Moreover, they are able to provide reasonable results also in cases when the assumption is not fulfilled. At the general level, it is good to start with those simple tools where the normality is assumed and see. If they do not work properly, to try something more sophisticated. If they do, to test the normality of the noise using some appropriate tests.
White noise is the denomination of a random variable that presents mean 0 and constant variance and is serially uncorrelated. White noise does not need to be normal distributed, but if it is, then you denominate as Gaussian White Noise. There are a bunch of tests to be done in a time series that you can obtain an answer showing if your series is a normal distribution or not. Example Jarque–Bera test or Shapiro–Wilk test.