Weak perturbations can be strongly enhanced in many-body systems that have dense spectra of excited states (compound nuclei, rare-earth atoms, molecules, clusters, quantum dots, etc.). Statistical consideration shows that in the case of zero-width states the probability distribution for the effect of the perturbation has an infinitte variance and does not obey the standard central limit theorem, i.e., the probability density for the average effect X=1/nΣi=1nxi does not tend to a Gaussian (normal) distribution with variance σn=σ1/√n , where n is the ‘‘number of measurements.’’ Since for probability densities of this form [f(x)≃a/x2 at large x] the central limit theorem is Fn(X)=a/X2+π2a2 at n≫1, the breadth of the distribution does not decrease with the increase of n. This means the following. (1) In spite of the random signs of observable effects for different compound states the probability of finding a large average effect for n levels is the same as that for a single-resonance measurements. (2) In some cases one does not need to resolve individual compound resonances and the enhanced value of the effect can be observed in the integral spectrum. This substantially increases the chances to observe statistical enhancement of weak perturbations in different reactions and systems. (3) The average value of parity and time-nonconserving effects in low-energy nucleon scattering cannot be described by a smooth weak optical potential. This ‘‘potential’’ would randomly fluctuate as a function of energy, with typical magnitudes much larger than the nucleon-nucleus weak potential. The effect of finite compound-state widths is considered.