one with positive energy escapes to infinity.

It would be useful to get more quantitative information on the spectral properties of the local observables, in particular estimates on negative fluctuations. One could try to make an explicit spectral resolution of these quantities, e.g. of the energy, contained in a finite volume, $V$, but this turns out to be difficult in general, even if one has given an explicit model theory in some Hilbert space. As we prefer a more general, model independent approach (not necessarily based on Hilbert space mathematics), we proceed by using (similar to Bell in his papers) a general probabilistic approach which rather exploits the statistics of individual measurement results. Unfortunately, we found that the standard estimates, known to us in this context (e.g. the Markov-Chebyshev-inequality), always go in the wrong direction (see e.g. [@Bauer] or [@Feller]). Therefore we present in the following our own estimate.

The strategy is the following. We take an observable, $E_V$, localized in $V$ with, for convenience, discrete spectral values, $E_i$, and corresponding probabilities denoted by $p_i>0$. If we assume that the expectation of $E_V$ is zero (which can always be achieved by a simple shift) we have $$\sum\,p_i=1\quad , \quad \sum\,p_i\cdot
  