  Thermodynamic stability of statistical systems requires that susceptibilities
be semipositive and finite. Susceptibilities are known to be related to the
fluctuations of extensive observable quantities. This relation becomes
nontrivial, when the operator of an observable quantity is represented as a sum
of operators corresponding to the extensive system parts. The association of
the dispersions of the partial operator terms with the total dispersion is
analyzed. A special attention is paid to the dependence of dispersions on the
total number of particles N in the thermodynamic limit. An operator dispersion
is called thermodynamically normal, if it is proportional to N at large values
of the latter. While, if the dispersion is proportional to a higher power of N,
it is termed thermodynamically anomalous. The following theorem is proved: The
global dispersion of a composite operator, which is a sum of linearly
independent self-adjoint terms, is thermodynamically anomalous if and only if
at least one of the partial dispersions is anomalous, the power of N in the
global dispersion being defined by the largest partial dispersion. Conversely,
the global dispersion is thermodynamically normal if and only if all partial
dispersions are normal. The application of the theorem is illustrated by
several examples of statistical systems. The notion of representative ensembles
is formulated. The relation between the stability and equivalence of
statistical ensembles is discussed.
