Let $a_1,a_2,\dots,a_n$ be independent identically distributed random vectors in $\mathbb R^n$. I need a bound for $E[|\det A|^{-1}]$, where $A$ is the matrix composed out of these vectors.

More specifically, these vectors take their values on a curve.

And more generally, I will be happy even if there is an estimate for a "non-square" determinant, precisely, for $E[G^{-1/2}]$, where $G$ is the Grammian determinant of $n$ iid vector in $\mathbb R^m$ and $m>n$.

Let $a_1,a_2,\dots,a_n$ be independent identically distributed random vectors in $\mathbb R^n$. I need a bound for $E[|\det A|^{-1}]$, where $A$ is the matrix composed out of these vectors.

More specifically, these vectors take their values on a curve.

And more generally, I will be happy even if there is an estimate for a "non-square" determinant, precisely, for $E[G^{-1/2}]$, where $G$ is the Grammian determinant of $n$ iid vector in $\mathbb R^m$ and $m>n$.

Update: yes, I need an upper bound.

Precisely, I have something like $a_i = (f_1(\xi_i),f_2(\xi_i),\dots,f_n(\xi_i))$, where $f_j(x) = |x|^{-\alpha_i}\sin(\beta_i x)$ with $\alpha_i\in(1,2)$ and look for an estimate in terms of $|\alpha_i-\alpha_j|$ and $|\beta_i-\beta_j|$.

I'm flexible with the choice of distribution for $\xi$, but it should be the same for all sets of $\alpha$'s and $\beta$'s.