Let $N>M$ and let $U$ be a $N\times M$ matrix of $iid$ random variables such that
$E\left[U\right]=0_{N\times M}$,
$Var\left[vec\:U\right]=I_{M}\otimes I_{N}$,
and, for each component $u_{i,l}$ of $U$ (where $i=1,\cdots,N$ and $l=1,\cdots,M$ ), $E\left[u_{i,l}^{3}\right]=\mu_{3}\neq0$ and have finite fourth moments. Now consider the matrix of real numbers $q\in\mathbb{R}^{N\times(N-M)}$ such that $q^{T}q=I_{\left(N-M\right)}$. I would like to establish when each component of $V=q^{T }U$ have finite fourth moments (even when $N$ and $M$ are going to infinite with $\frac{M}{N}\rightarrow\alpha\in(0,1)$). Any suggestion?