# How to tell is time series is Gaussian

For the Gaussian process, this is the definition provided in class: A process is Gaussian if all the finite-dimensional distributions have a normal distribution.

This means that $$[X_{t_1}, . . . , X_{t_k}]'∼ Nk (m, Σ)$$ where $$m'= [µ_X(t1) µ_X(t2) · · · µ_X(tk)]$$ and the $$(r,s)$$th entry of the $$k × k$$ matrix $$Σ$$ is $$γ_X(r,s)$$. Therefore a Gaussian process can be identified by its mean function and its autocovariance function (or autocorrelation function). Therefore imposing properties on $$m$$ and $$Σ$$ will be reflected uniquely on the process.

Gaussian is important to understand, and we use the properties of it:

1. Gaussian time series is strict stationary is equivalent to weak stationary.
2. $$X,Y$$ are independent random variables iff $$X,Y$$ are uncorrelated.
3. If the process is Gaussian, this means that the mean function is free from $$h$$ and the autocovariance function $$γ_X(r,s)$$ only depends on $$|r − s|$$.

Now here's what I'm having a hard time with:

• The definition where we say "Therefore a Gaussian process can be identified by its mean function and its autocovariance function (or autocorrelation function). Therefore imposing properties on $$m$$ and $$Σ$$ will be reflected uniquely on the process."

I'm not understanding what exactly it means when it says we can identify the Gaussian process by the mean function and autocovariance function. And what properties are imposing on $$m$$ and $$\sum$$.

• How do you know is a time series $$\{Y_t\}$$ is Gaussian?

• Is white noise considered to be Gaussian?

I'm a bit confused over this, because I thought white noise can be any sequence of independent random variables $${Z_t : t=0,1,2,..}$$ where $$Z_t$$ is iid and it's WN when $$E[Z_t]=0$$ and $$Var(Z_t)=\sigma^2 < \infty$$.

Does this somehow imply WN is Gaussian?

• Is $$MA(q)$$ process gaussian?

The reason I ask if WN or $$MA(q)$$ implies Gaussian process is because of problem 1.5 in Brockwell and Davis Time Series Analysis textbook. http://home.iitj.ac.in/~parmod/document/introduction%20time%20series.pdf

Part b) of 1.5 was solved in class using $$Cov(X_r,X_s) = \gamma(|r-s|)$$. This means we assume $$X_t$$ is a Gaussian process. But how do we know it is Gaussian? And why do we need the covariance to calculate the variance of the sample mean?

• From $\mathbb E[X_t]$ and $\operatorname{Cov}(X_t,X_s)$ alone one cannot determine if $X$ is Gaussian. In the setting you found this quote I am sure they consider only Gaussian processes $X$ and nothing else. In this answer you can see a few examples of imporatant Gaussian processes which are all characterized by their expectation and covariance function. Feb 22 at 16:36