On p×1p×1 dependent random variables having each (p−1)×1(p−1)×1 sub-vector made up of IID observations with examples

It is not easy to locate a non-NpNp joint probability density function (p.d.f.) such that each (p−1)(p−1)-dimensional sub-vector would consist of p−1p−1 independent and identically distributed (i.i.d.) standard normal variables, but the construction of such a multivariate distribution can be interesting. We address this problem and provide examples of this and other kinds including joint p.d.f.s where each (p−1)(p−1)-dimensional sub-vector consists of p−1p−1 i.i.d. normal, Laplace or lognormal variables.