The OP needs some correction:
The title is "Gaussian Processes". Okay, so, we're considering "processes". A Gaussian process is a special case of a stochastic process.
Here is the usual, first definition of a stochastic process: For the set of real numbers R and for each t in R (t usually regarded as time) we have a random variable X_t (TeX notation for subscripts) where the values of X_t are in R. That is, X_t is a real valued random variable. So, the stochastic process is, say,
{X_t | t in R}
This stochastic process is a Gaussian process if for each t in R X_t has Gaussian distribution.
Commonly we continue and assume that both the expectation and variance of each X_t is the same for each t in R.
In addition for any s in R we might assume that the covariance
Cov(X_t, X_{s + t})
(more TeX notation) is a constant for all t.
With these assumptions we can do a lot in power spectral estimation and extrapolation.
We can apply the central limit theorem to argue that an independent increments process is Gaussian.
For more, see, e.g.,
I. A. Ibragimov and Y. A. Rozanov,
Gaussian Random Processes,
ISBN 0-387-90302-x,
Springer-Verlag,
New York,
1978.
For one more, if real valued random variables X and Y are independent, then they are uncorrelated. Net, independence is a much stronger assumption than uncorrelated.
But, amazingly, if X and Y are joint Gaussian and uncorrelated, then they are independent.
For one more, for the example of curve fitting and interpolation, I'd suggest not polynomials but splines or least squares splines.
For one more, for Gaussian X_t for finitely many t, the sample mean and sample variance are sufficient statistics. That means that in any work in statistics, we can keep the sample mean and variance, throw away the X_t, and in principle do as well. The foundation paper on sufficient statistics was by Halmos and Savage back in the 1940s. The concept of sufficient statistics is an application of the Radon-Nikodym theorem. E. Dynkin showed that the Gaussian assumption is critical -- with small changes the result is false.
The title is "Gaussian Processes". Okay, so, we're considering "processes". A Gaussian process is a special case of a stochastic process.
Here is the usual, first definition of a stochastic process: For the set of real numbers R and for each t in R (t usually regarded as time) we have a random variable X_t (TeX notation for subscripts) where the values of X_t are in R. That is, X_t is a real valued random variable. So, the stochastic process is, say,
{X_t | t in R}
This stochastic process is a Gaussian process if for each t in R X_t has Gaussian distribution.
Commonly we continue and assume that both the expectation and variance of each X_t is the same for each t in R.
In addition for any s in R we might assume that the covariance
Cov(X_t, X_{s + t})
(more TeX notation) is a constant for all t.
With these assumptions we can do a lot in power spectral estimation and extrapolation.
We can apply the central limit theorem to argue that an independent increments process is Gaussian.
For more, see, e.g.,
I. A. Ibragimov and Y. A. Rozanov, Gaussian Random Processes, ISBN 0-387-90302-x, Springer-Verlag, New York, 1978.
For one more, if real valued random variables X and Y are independent, then they are uncorrelated. Net, independence is a much stronger assumption than uncorrelated.
But, amazingly, if X and Y are joint Gaussian and uncorrelated, then they are independent.
For one more, for the example of curve fitting and interpolation, I'd suggest not polynomials but splines or least squares splines.
For one more, for Gaussian X_t for finitely many t, the sample mean and sample variance are sufficient statistics. That means that in any work in statistics, we can keep the sample mean and variance, throw away the X_t, and in principle do as well. The foundation paper on sufficient statistics was by Halmos and Savage back in the 1940s. The concept of sufficient statistics is an application of the Radon-Nikodym theorem. E. Dynkin showed that the Gaussian assumption is critical -- with small changes the result is false.