I was conducting the Karhunen-Loeve (K-L) Expansion for a random vector.
Based on the KL expansion, I transformed the original random vector into a standardized random vector $\boldsymbol{X}=[X_1,X_2,\dots,X_n]$ with zero mean and identity covariance. Now I am wondering whether there are some relationships I can get for the coskewness and cokurtosis, i.e., $\mathbb{E}(X_iX_jX_k)$ and $\mathbb{E}(X_jX_jX_kX_l),\;\forall i,j,k,l$. Do these values become very small or zero when the random variables are uncorrelated?
Anything inspiring will be appreciated. Thank you very much in advance.