interpretation as (a square of) the projection of the vector $\sigma^{-1}\mathbf{V}_p$ along the direction given by the vector $\mathbf{V}_n$. These projections are statistically independent, as is obvious from the above discussion. The probability distribution of any projection follows readily from Eqs. (\[pt\]) and (\[pg\]). Performing an integration first over $\gamma$ and then over $\theta$ in the definition $P(Z)={\left \langle \delta(Z-\gamma\cos^2\theta) \right \rangle}$, one finds $$\label{pZ}
 P(Z)=\frac{1}{\sqrt{2\pi Z}}e^{-Z/2}\,.$$ Thus, surprisingly, the distribution of $Z_p$ is independent of $M$, being given by the Porter-Thomas law at any $M\geq1$.

Average of $X$ and width fluctuations
-------------------------------------

A general expression of the average value of the complexness parameter $X$ can be readily found from Eq. (\[X\]) by making use of the mutual statistical independence between the widths $\{\gamma_n\}$, the projections $\{Z_n\}$ and the levels $\{E_n\}$. Noting that ${\left \langle \gamma \right \rangle}=M$ and ${\left \langle Z \right \rangle}=1={\left \langle \gamma \right \rangle}{\left \langle \cos^2\theta \right \rangle}$, one obtains $$\label{Xav}
  {\left \langle X \right \rangle} = Mf \,,$$ where the factor $f$ depends on the statistical properties of the energies of the closed system only, $$\label{f}
    f = {\left \langle \sum_{p\ne n}\frac{\Delta^2}{4(E_n-E_p)^2} \right \rangle}.$$

It is important to note that, generally, the nonzero values of 