  We compute statistical properties of the lowest-order multipole coefficients
of the magnetic field generated by a dynamo of arbitrary shape. To this end we
expand the field in a complete biorthogonal set of base functions, viz. B =
sum_k a^k(t) b^k(r). We consider a linear problem and the statistical
properties of the fluid flow are supposed to be given. The turbulent convection
may have an arbitrary distribution of spatial scales. The time evolution of the
expansion coefficients a^k(t) is governed by a stochastic differential equation
from which we infer their averages <a^k>, autocorrelation functions <a^k(t)
a^{k*}(t+tau)>, and an equation for the cross correlations <a^k a^l*>. The
eigenfunctions of the dynamo equation (with eigenvalues lambda_k) turn out to
be a preferred set in terms of which our results assume their simplest form.
The magnetic field of the dynamo is shown to consist of transiently excited
eigenmodes whose frequency and coherence time is given by Im(lambda_k) and
-1/(Re lambda_k), respectively. The relative r.m.s. excitation level of the
eigenmodes, and hence the distribution of magnetic energy over spatial scales,
is determined by linear theory. An expression is derived for <|a^k|^2> /
<|a^0|^2> in case the fundamental mode b^0 has a dominant amplitude, and we
outline how this expression may be evaluated. It is estimated that
<|a^k|^2>/<|a^0|^2> ~ 1/N where N is the number of convective cells in the
dynamo. We show that the old problem of a short correlation time (or FOSA) has
been partially eliminated. Finally we prove that for a simple statistically
steady dynamo with finite resistivity all eigenvalues obey Re(lambda_k) < 0.
