a sum of correlated random variables which, therefore, does not obey the standard central limit theorem. Statistics of a similar kind of objects appears, e.g., in the study of the parametric level dynamics (“curvature”) [@vonOppen1995] and in the context of interference effects in neutron scattering from compound nucleus [@Ber00].

Rescaled parameters and their statistics
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The complexness factor (\[q2def\]) contains two contributions of distinct types, one is due to the internal levels and the other is due to the coupling matrix elements $\Gamma_{np}$. From a statistical point of view, these two are statistically independent of one another. We note, however, that the levels $E_n$ are mutually correlated. The quantities $\Gamma_{np}$’s, unlike the original amplitudes $V_n^c$, are also not statistically independent. Although their joint distribution can be found from (\[V\]), the resulting expression is quite complicated [@Yu02], being of little practical use for actual calculations in the present context.

To overcome the difficulty of averaging over the coupling amplitudes, we follow Sokolov and Zelevinsky [@Sok89] and treat an arbitrary matrix element $\Gamma_{np}$ as a scalar products between $M$-dimensional vectors $\mathbf{V}_n$ and $\mathbf{V}_p$ of the coupling amplitudes $\{V_l^c\}$ associated with the levels $l=n$ and $l=p$. This suggests a natural parametrization for $\Gamma_{np}$ 